Hypoid gear design method and hypoid gear

ABSTRACT

A degree of freedom of a hypoid gear is improved. Instantaneous axis in a relative rotation of a gear axis and a pinion axis, a line of centers, intersection between the instantaneous axis and the line of centers, and an inclination angle of the instantaneous axis regarding the rotation axis of the gear are calculated based on a shaft angle, an offset, and a gear ratio of a hypoid gear. Based on these variables, base coordinate systems are determined, and the specifications are calculated using these coordinate systems. For spiral angles, pitch cone angles, and reference circle radii of the gear and pinion, one of the values for the gear and the pinion is set and a design reference point calculated. Based on the design reference point and a contact normal of the gear, specifications are calculated. The pitch cone angle of the gear or the pinion can be selected.

The present application is a divisional application of U.S. applicationSer. No. 13/054,323, filed Feb. 3, 2011, which is a National Stage Entryof PCT/JP2009/063234, filed Jul. 16, 2009, which claims priority to eachof JP 2009-111881, filed May 1, 2009, JP 2008-280558, filed Oct. 30,2008, and JP 2008-187965, filed Jul. 18, 2008. The disclosures of eachof the above applications are hereby incorporated by reference in theirentireties.

TECHNICAL FIELD

The present invention relates to a method of designing a hypoid gear.

BACKGROUND ART

A design method of a hypoid gear is described in Ernest Wildhaber, BasicRelationship of Hypoid Gears, American Machinist, USA, Feb. 14, 1946, p.108-111 and in Ernest Wildhaber, Basic Relationship of Hypoid Gears II,American Machinist, USA, Feb. 28, 1946, p. 131-134. In these references,a system of eight equations is set and solved (for cone specificationsthat contact each other) by setting a spiral angle of a pinion and anequation of a radius of curvature of a tooth trace, in order to solveseven equations with nine variables which are obtained by setting, asdesign conditions, a shaft angle, an offset, a number of teeth, and aring gear radius. Because of this, the cone specifications such as thepitch cone angle Γ_(gw) depend on the radius of curvature of the toothtrace, and cannot be arbitrarily determined.

In addition, in the theory of gears in the related art, a tooth trace isdefined as “an intersection between a tooth surface and a pitchsurface”. However, in the theory of the related art, there is no commongeometric definition of a pitch surface for all kinds of gears.Therefore, there is no common definition of the tooth trace and ofcontact ratio of the tooth trace for various gears from cylindricalgears to hypoid gears. In particular, in gears other than thecylindrical gear and a bevel gear, the tooth trace is not clear.

In the related art, the contact ratio m_(f) of tooth trace is defined bythe following equation for all gears.

m _(f) =F tan ψ₀ /p

where, p represents the circular pitch, F represents an effective facewidth, and ψ₀ represents a spiral angle.

Table 1 shows an example calculation of a hypoid gear according to theGleason method. As shown in this example, in the Gleason design method,the tooth trace contact ratios are equal for a drive-side tooth surfaceand for a coast-side tooth surface. This can be expected because of thecalculation of the spiral angle ψ₀ as a virtual spiral bevel gear withψ₀=(ψ_(pw)+ψ_(gw))/2 (refer to FIG. 9).

The present inventors, on the other hand, proposed in Japanese PatentNo. 3484879 a method for uniformly describing the tooth surface of apair of gears. In other word, a method for describing a tooth surfacehas been shown which can uniformly be used in various situations from apair of gears having parallel axes, which is the most widely usedconfiguration, to a pair of gears whose axes do not intersect and arenot parallel with each other (skew position).

There is a desire to determine the cone specifications independent fromthe radius of curvature of the tooth trace, and to increase the degreeof freedom of the design.

In addition, in a hypoid gear, the contact ratio and the transmissionerror based on the calculation method of the related art are notnecessarily correlated to each other. Of the contact ratios of therelated art, the tooth trace contact ratio has the same value betweenthe drive-side and the coast-side, and thus the theoretical basis isbrought into question.

An advantage of the present invention is that a hypoid gear designmethod is provided which uses the uniform describing method of the toothsurface described in JP 3484879, and which has a high degree of freedomof design.

Another advantage of the present invention is that a hypoid gear designmethod is provided in which a design reference body of revolution (pitchsurface) which can be applied to the hypoid gear, the tooth trace, andthe tooth trace contact ratio are newly defined using the uniformdescribing method of the tooth surface described in JP 3484879, and thenewly defined tooth trace contact ratio is set as a design index.

SUMMARY OF THE INVENTION

According to one aspect of the present invention, there is provided adesign method of a hypoid gear wherein an instantaneous axis S which isan axis of a relative angular velocity of a first gear and a secondgear, a line of centers v_(c) which is common to rotation axes of thefirst gear and the second gear, an intersection C_(s) between theinstantaneous axis S and the line of centers v_(c), and an inclinationangle Γ_(S) of the instantaneous axis S with respect to the rotationaxis of the second gear are calculated based on a shaft angle Σ, anoffset E, and a gear ratio i₀ of a hypoid gear, basic coordinate systemsC₁, C₂, and C_(s) are determined from these variables, andspecifications are calculated based on the coordinate systems. Inparticular, the specifications are calculated by setting a common pointof contact of pitch cones of the first gear and the second gear as adesign reference point P_(w).

When an arbitrary point (design reference point) P_(w) is set in astatic space, six cone specifications which are in contact at the pointP_(w) are represented by coordinates (u_(cw), v_(cw), z_(cw)) of thepoint P_(w) based on a plane (pitch plane) S_(t) defined by a peripheralvelocity V_(1w) and a peripheral velocity V_(2w) at the point P_(w) anda relative velocity V_(rsw). Here, the cone specifications refer toreference circle radii R_(1w) and R_(2w) of the first gear and thesecond gear, spiral angles ψ_(pw) and ψ_(gw) of the first gear and thesecond gear, and pitch cone angles γ_(pw) and Γ_(gw) of the first gearand the second gear. When three of these cone specifications are set,the point P_(w) is set, and thus the remaining three variables are alsoset. In other words, in various aspects of the present invention, thespecifications of cones which contact each other are determined basedmerely on the position of the point P_(w) regardless of the radius ofcurvature of the tooth trace.

Therefore, it is possible to set a predetermined performance as a designtarget function, and select the cone specifications which satisfy thetarget function with a high degree of freedom. Examples of the designtarget function include, for example, a sliding speed of the toothsurface, strength of the tooth, and the contact ratio. The performancerelated to the design target function is calculated while the conespecification, for example the pitch cone angle Γ_(gw), is changed, andthe cone specification is changed and a suitable value is selected whichsatisfies the design request.

According to one aspect of the present invention, an contact ratio isemployed as the design target function, and there is provided a methodof designing a hypoid gear wherein a pitch cone angle Γ_(gcone) of onegear is set, an contact ratio is calculated, the pitch cone angleΓ_(gcone) is changed so that the contact ratio becomes a predeterminedvalue, a pitch cone angle Γ_(gw) is determined, and specifications arecalculated based on the determined pitch cone angle Γ_(gw). As describedabove, the contact ratio calculated by the method of the related artdoes not have a theoretical basis. In this aspect of the presentinvention, a newly defined tooth trace and an contact ratio related tothe tooth trace are calculated, to determine the pitch cone angle. Thetooth surface around a point of contact is approximated by itstangential plane, and a path of contact is made coincident to anintersection of the surface of action (pitch generating line L_(pw)),and a tooth trace is defined as a curve on a pitch hyperboloid obtainedby transforming the path of contact into a coordinate system whichrotates with each gear. Based on the tooth trace of this new definition,the original contact ratio of the hypoid gear is calculated and thecontact ratio can be used as an index for design. A characteristic ofthe present invention is in the definition of the pitch cone anglerelated to the newly defined tooth trace.

According to another aspect of the present invention, it is preferablethat, in the hypoid gear design method, the tooth trace contact ratio isassumed to be 2.0 or more in order to achieve constant engagement of twogears with two or more teeth.

When the pitch cone angle Γ_(gw) is set to an inclination angle Γ_(s) ofan instantaneous axis, the contact ratios of the drive-side and thecoast-side can be set approximately equal to each other. Therefore, itis preferable for the pitch cone angle to be set near the inclinationangle of the instantaneous axis. In addition, it is also preferable toincrease one of the contact ratios of the drive-side or coast-side asrequired. In this process, first, the pitch cone angle is set at theinclination angle of the instantaneous axis and the contact ratio iscalculated, and a suitable value is selected by changing the pitch coneangle while observing the contact ratio. It is preferable that a widthof the change of the pitch cone angle be in a range of ±5° with respectto the inclination angle Γ_(s) of the instantaneous axis. This isbecause if the change is out of this range, the contact ratio of one ofthe drive-side and the coast-side will be significantly reduced.

More specifically, according to one aspect of the present invention, ahypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of ahypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gearratio i₀, an instantaneous axis S which is an axis of a relative angularvelocity of a first gear and a second gear, a line of centers v_(c) withrespect to rotation axes of the first gear and the second gear, anintersection C_(s) between the instantaneous axis S and the line ofcenters v_(c), and an inclination angle Γ_(s) of the instantaneous axisS with respect to the rotation axis of the second gear, and determiningcoordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radiusR_(1w) of the first gear and a reference circle radius R_(2w) of thesecond gear, one of a spiral angle ψ_(pw) of the first gear and a spiralangle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) ofthe first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a commonpoint of contact of pitch cones of the first gear and the second gear,and the three other variables which are not set in the step (c), basedon the three variables which are set in the step (c);

(e) setting a contact normal g_(wD) of a drive-side tooth surface of thesecond gear;

(f) setting a contact normal g_(wC) of a coast-side tooth surface of thesecond gear; and

(g) calculating specifications of the hypoid gear based on the designreference point P_(w), the three variables which are set in the step(c), the contact normal g_(wD) of the drive-side tooth surface of thesecond gear, and the contact normal g_(wC) of the coast-side toothsurface of the second gear.

According to another aspect of the present invention, a hypoid gear isdesigned according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of ahypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gearratio i₀, an instantaneous axis S which is an axis of a relative angularvelocity of a first gear and a second gear, a line of centers v_(c) withrespect to rotation axes of the first gear and the second gear, anintersection C_(s) between the instantaneous axis S and the line ofcenters v_(c), and an inclination angle Γ_(s) of the instantaneous axisS with respect to the rotation axis of the second gear, and determiningcoordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radiusR_(1w) of the first gear and a reference circle radius R_(2w) of thesecond gear, one of a spiral angle ψ_(pw) of the first gear and a spiralangle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) ofthe first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a commonpoint of contact of pitch cones of the first gear and the second gear,and the three other variables which are not set in the step (c), basedon the three variable which are set in the step (c);

(e) calculating a pitch generating line L_(pw) which passes through thedesign reference point P_(w) and which is parallel to the instantaneousaxis S;

(f) setting an internal circle radius R_(2t) and an external circleradius R_(2h) of the second gear;

(g) setting a contact normal g_(wD) of a drive-side tooth surface of thesecond gear;

(h) calculating an intersection P_(0D) between a reference plane S_(H)which is a plane orthogonal to the line of centers v_(c) and passingthrough the intersection C_(s) and the contact normal g_(wD) and aradius R_(20D) of the intersection P_(0D) around a gear axis;

(i) calculating an inclination angle φ_(s0D) of a surface of actionS_(wD) which is a plane defined by the pitch generating line L_(pw) andthe contact normal g_(wD) with respect to the line of centers v_(c), aninclination angle ψ_(sw0D) of the contact normal g_(wD) on the surfaceof action S_(wD) with respect to the instantaneous axis S, and one pitchP_(gwD) on the contact normal g_(wD);

(j) setting a provisional second gear pitch cone angle Γ_(gcone), andcalculating an contact ratio m_(fconeD) of the drive-side tooth surfacebased on the internal circle radius R_(2t) and the external circleradius R_(2h);

(k) setting a contact normal g_(wC) of a coast-side tooth surface of thesecond gear;

(l) calculating an intersection P_(0C) between the reference plane S_(H)which is a plane orthogonal to the line of centers v_(c) and passingthrough the intersection C_(s) and the contact normal g_(wC) and aradius R_(20c) of the intersection P_(0C) around the gear axis;

(m) calculating an inclination angle φ_(s0C) of a surface of actionS_(wC) which is a plane defined by the pitch generating line L_(pw) andthe contact normal g_(wC) with respect to the line of centers v_(c), aninclination angle ψ_(sw0C) of the contact normal g_(wC) on the surfaceof action S_(wC) with respect to the instantaneous axis S, and one pitchP_(gwC) on the contact normal g_(wC);

(n) setting a provisional second gear pitch cone angle Γ_(gcone), andcalculating an contact ratio m_(fconeC) of the coast-side tooth surfacebased on the internal circle radius R_(2t) and the external circleradius R_(2h);

(o) comparing the contact ratio m_(fconeD) of the drive-side toothsurface and the contact ratio m_(fconeC) of the coast-side toothsurface, and determining whether or not these contact ratios arepredetermined values;

(p) when the contact ratios of the drive-side and the coast-side are thepredetermined values, replacing the provisional second gear pitch coneangle Γ_(gcone) with the second gear pitch cone angle Γ_(gw) obtained inthe step (c) or in the step (d);

(q) when the contact ratios of the drive-side and the coast-side are notthe predetermined values, changing the provisional second gear pitchcone angle Γ_(gcone) and re-executing from step (g);

(r) re-determining the design reference point P_(w), the other one ofthe reference circle radius R_(1w) of the first gear and the referencecircle radius R_(2w) of the second gear which is not set in the step(c), the other one of the spiral angle ψ_(pw) of the first gear and thespiral angle ψ_(gw) of the second gear which is not set in the step (c),and the first gear pitch cone angle γ_(pw) based on the one of thereference circle radius R_(1w) of the first gear and the referencecircle radius R_(2w) of the second gear which is set in the step (c),the one of the spiral angle ψ_(pw) of the first gear and the spiralangle ψ_(gw) of the second gear which is set in the step (c), and thesecond gear pitch cone angle Γ_(gw) which is replaced in the step (p),and

(s) calculating specifications of the hypoid gear based on thespecifications which are set in the step (c), the specifications whichare re-determined in the step (r), the contact normal g_(wD) of thedrive-side tooth surface of the second gear, and the contact normalg_(wC) of the coast-side tooth surface of the second gear.

According to another aspect of the present invention, a hypoid gear isdesigned according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of ahypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gearratio i₀, an instantaneous axis S which is an axis of a relative angularvelocity of a first gear and a second gear, a line of centers v_(c) withrespect to rotation axes of the first gear and the second gear, anintersection C_(s) between the instantaneous axis S and the line ofcenters v_(c), and an inclination angle Γ_(s) of the instantaneous axisS with respect to the rotation axis of the second gear, to determinecoordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting three variables including one of a reference circle radiusR_(1w) of the first gear and a reference circle radius R_(2w) of thesecond gear, one of a spiral angle ψ_(pw) of the first gear and a spiralangle ψ_(gw) of the second gear, and one of a pitch cone angle γ_(pw) ofthe first gear and a pitch cone angle Γ_(gw) of the second gear;

(d) calculating the design reference point P_(w), which is a commonpoint of contact of pitch cones of the first gear and the second gear,and the three other variables which are not set in the step (c), basedon the three variables which are set in the step (c);

(e) calculating a pitch generating line L_(pw) which passes through thedesign reference point P_(w) and which is parallel to the instantaneousaxis S;

(f) setting an internal circle radius R_(2t) and an external circleradius R_(2h) of the second gear;

(g) setting a contact normal g_(wD) of a drive-side tooth surface of thesecond gear;

(h) calculating an intersection P_(0D) between a reference plane S_(H)which is a plane orthogonal to the line of centers v_(c) and passingthrough the intersection C_(s) and the contact normal g_(wD) and aradius R_(20D) of the intersection P_(0D) around a gear axis;

(i) calculating an inclination angle φ_(a0D) of a surface of actionS_(wD) which is a plane defined by the pitch generating line L_(pw) andthe contact normal g_(wD) with respect to the line of centers v_(c), aninclination angle ψ_(sw0D) of the contact normal g_(wD) on the surfaceof action S_(wD) with respect to the instantaneous axis S, and one pitchP_(gwD) on the contact normal g_(wD);

(j) setting a provisional second gear pitch cone angle Γ_(gcone), andcalculating an contact ratio m_(fconeD) of the drive-side tooth surfacebased on the internal circle radius R_(2t) and the external circleradius R_(2h);

(k) setting a contact normal g_(wC) of a coast-side tooth surface of thesecond gear;

(l) calculating an intersection P_(0c) between the reference plane S_(H)which is a plane orthogonal to the line of centers v_(c) and passingthrough the intersection C_(s) and the contact normal g_(wC) and aradius R_(20c) of the intersection P_(0c) around the gear axis;

(m) calculating an inclination angle φ_(s0C) of a surface of actionS_(wC) which is a plane defined by the pitch generating line L_(pw) andthe contact normal g_(wC) with respect to the line of centers v_(c), aninclination angle ψ_(sw0C) of the contact normal g_(wC) on the surfaceof action S_(wC) with respect to the instantaneous axis S, and one pitchP_(gwC) on the contact normal g_(wC);

(n) setting a provisional second gear pitch cone angle Γ_(gcone), andcalculating an contact ratio m_(fconeC) of the coast-side tooth fconeCsurface based on the internal circle radius R_(2t) and the externalcircle radius R_(2h);

(o) comparing the contact ratio m_(fconeD) of the drive-side toothsurface and the contact ratio m_(fconeC) of the coast-side toothsurface, and determining whether or not these contact ratios arepredetermined values;

(p) changing, when the contact ratios of the drive-side and thecoast-side are not the predetermined values, the provisional second gearpitch cone angle Γ_(gcone) and re-executing from step (g);

(q) defining, when the contact ratios of the drive-side and thecoast-side are the predetermined values, a virtual cone having theprovisional second gear pitch cone angle Γ_(gcone) as a cone angle;

(r) calculating a provisional pitch cone angle γ_(pcone) of the virtualcone of the first gear based on the determined pitch cone angleΓ_(gcone); and

(s) calculating specifications of the hypoid gear based on the designreference point P_(w), the reference circle radius R_(1w) of the firstgear and the reference circle radius R_(2w) of the second gear which areset in the step (c) and the step (d), the spiral angle ψ_(pw) of thefirst gear and the spiral angle ψ_(gw) of the second gear which are setin the step (c) and the step (d), the cone angle Γ_(gcone) of thevirtual cone and the cone angle γ_(pcone) of the virtual cone which aredefined in the step (q) and the step (r), the contact normal g_(wD) ofthe drive-side tooth surface of the second gear, and the contact normalg_(wC) of the coast-side tooth surface of the second gear.

According to another aspect of the present invention, in a method ofdesigning a hypoid gear, a pitch cone angle of one gear is set equal toan inclination angle of an instantaneous axis, and the specificationsare calculated. When the pitch cone angle is set equal to theinclination angle of the instantaneous axis, the contact ratios of thedrive-side tooth surface and the coast-side tooth surface become almostequal to each other. Therefore, a method is provided in which the pitchcone angle is set to the inclination angle of the instantaneous axis ina simple method, that is, without reviewing the contact ratios indetail.

More specifically, according to another aspect of the present invention,the hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of ahypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gearratio i₀, an instantaneous axis S which is an axis of a relative angularvelocity of a first gear and a second gear, a line of centers v_(c) withrespect to rotational axes of the first gear and the second gear, anintersection C_(s) between the instantaneous axis S and the line ofcenters v_(c), and an inclination angle Γ_(s) of the instantaneous axiswith respect to the rotational axis of the second gear;

(c) determining the inclination angle Γ_(s) of the instantaneous axis asa second gear pitch cone angle Γ_(gw); and

(d) calculating specifications of the hypoid gear based on thedetermined second gear pitch cone angle Γ_(gw). According to anotheraspect of the present invention, in a method of designing a hypoid gear,a design reference point P_(w) is not set as a point of contact betweenthe pitch cones of the first gear and second gear, but is determinedbased on one of reference circle radii R_(1w) and R_(2w) of the firstgear and the second gear, a spiral angle ψ_(rw), and a phase angle β_(w)of the design reference point, and the specifications are calculated.

More specifically, according to another aspect of the present invention,a hypoid gear is designed according to the following steps:

(a) setting a shaft angle Σ, an offset E, and a gear ratio i₀ of ahypoid gear;

(b) calculating, based on the shaft angle Σ, the offset E, and the gearratio i₀, an instantaneous axis S which is an axis of a relative angularvelocity of a first gear and a second gear, a line of centers v_(c) withrespect to rotation axes of the first gear and the second gear, anintersection C_(s) between the instantaneous axis S and the line ofcenters v_(c), and an inclination angle Γ_(s) of the instantaneous axisS with respect to the rotation axis of the second gear, to determinecoordinate systems C₁, C₂, and C_(s) for calculation of specifications;

(c) setting one of a reference circle radius R_(1w) of the first gearand a reference circle radius R_(2w) of the second gear, a spiral angleψ_(rw), and a phase angle β_(w) of a design reference point P_(w), todetermine the design reference point;

(d) calculating the design reference point P_(w) and a reference circleradius which is not set in the step (c) from a condition where the firstgear and the second gear share the design reference point P_(w), basedon the three variables which are set in the step (c);

(e) setting one of a reference cone angle γ_(pw) of the first gear and areference cone angle Γ_(gw) of the second gear;

(f) calculating a reference cone angle which is not set in the step (e),based on the shaft angle Σ and the reference cone angle which is set inthe step (e);

(g) setting a contact normal g_(wD) of a drive-side tooth surface of thesecond gear;

(h) setting a contact normal g_(wC) of a coast-side tooth surface of thesecond gear; and

(i) calculating specifications of the hypoid gear based on the designreference point P_(w), the reference circle radii R_(1w) and R_(2w), andthe spiral angle ψ_(rw) which are set in the step (c) and the step (d),the reference cone angles γ_(pw) and Γ_(gw) which are set in the step(e) and the step (f), and the contact normals g_(wC) and g_(wD) whichare set in the step (g) and the step (h).

The designing steps of these two aspects of the present invention can beexecuted by a computer by describing the steps with a predeterminedcomputer program. A unit which receives the gear specifications andvariables is connected to the computer and a unit which provides adesign result or a calculation result at an intermediate stage is alsoconnected to the computer.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a diagram showing external appearance of a hypoid gear.

FIG. 1B is a diagram showing a cross sectional shape of a gear.

FIG. 1C is a diagram showing a cross sectional shape of a pinion.

FIG. 2 is a diagram schematically showing appearance of coordinate axesin each coordinate system, a tooth surface of a gear, a tooth profile,and a path of contact.

FIG. 3 is a diagram showing a design reference point P₀ and a path ofcontact g₀ for explaining a variable determining method, with coordinatesystems C₂′, O_(q2) C₁′, and O_(q1).

FIG. 4 is a diagram for explaining a relationship between gear axes Iand II and an instantaneous axis S.

FIG. 5 is a diagram showing a relative velocity V_(s) at a point C_(s).

FIG. 6 is a diagram showing, along with planes S_(H), S_(s), S_(p), andS_(n), a reference point P₀, a relative velocity V_(rs0), and a path ofcontact g₀.

FIG. 7 is a diagram showing a relationship between a relative velocityV_(rs) and a path of contact g₀ at a point P.

FIG. 8 is a diagram showing a relative velocity V_(rs0) and a path ofcontact g₀ at a reference point P₀, with a coordinate system C_(s).

FIG. 9 is a diagram showing coordinate systems C₁, C₂, and C_(s) of ahypoid gear and a pitch generating line L_(pw).

FIG. 10 is a diagram showing a tangential cylinder of a relativevelocity V_(rsw).

FIG. 11 is a diagram showing a relationship between a pitch generatingline L_(pw), a path of contact g_(w), and a surface of action S_(w) at adesign reference point P_(w).

FIG. 12 is a diagram showing a surface of action using coordinatesystems C_(s), C₁, and C₂ in the cases of a cylindrical gear and acrossed helical gear.

FIG. 13 is a diagram showing a surface of action using coordinatesystems C_(s), C₁, and C₂ in the cases of a bevel gear and a hypoidgear.

FIG. 14 is a diagram showing a relationship between a contact pointP_(w) and points O_(1nw) and O_(2nw).

FIG. 15 is a diagram showing a contact point P_(w) and a path of contactg_(w) in planes S_(tw), S_(nw), and G_(2w).

FIG. 16 is a diagram showing a transmission error of a hypoid gearmanufactured as prototype that uses current design method.

FIG. 17 is an explanatory diagram of a virtual pitch cone.

FIG. 18 is a diagram showing a definition of a ring gear shape.

FIG. 19 is a diagram showing a definition of a ring gear shape.

FIG. 20 is a diagram showing a state in which an addendum is extendedand a tip cone angle is changed.

FIG. 21 is a diagram showing a transmission error of a hypoid gearmanufactured as prototype using a design method of a preferredembodiment of the present invention.

FIG. 22 is a schematic structural diagram of a system which aids adesign method for a hypoid gear.

FIG. 23 is a diagram showing a relationship between a gear ratio and atip cone angle of a uniform tooth depth hypoid gear designed using acurrent method.

FIG. 24 is a diagram showing a relationship between a gear ratio and atip cone angle of a tapered tooth depth hypoid gear designed using acurrent method.

FIG. 25 is a diagram showing a relationship between a tooth trace curveand a cutter radius of a uniform-depth tooth.

FIG. 26 is a diagram showing a relationship between a tooth trace curveand a cutter radius of a tapered-depth tooth.

DESCRIPTION OF THE PREFERRED EMBODIMENT

A preferred embodiment of the present invention will now be describedwith reference to the drawings.

1. Coordinate System of Hypoid Gear 1.1 Coordinate Systems C₁, C₂,C_(q1), and C_(q2)

In the following description, a small diameter gear in a pair of hypoidgears is referred to as a pinion, and a large diameter gear is referredto as a ring gear. In addition, in the following, the descriptions maybe based on the tooth surface, tooth trace, etc. of the ring gear, butbecause the pinion and the ring gear are basically equivalent, thedescription may similarly be based on the pinion. FIG. 1A is aperspective view showing external appearance of a hypoid gear. Thehypoid gear is a pair of gears in which a rotational axis (pinion axis)I of the pinion 10 and the rotational axis (gear axis) II of the ringgear 14 are not parallel and do not intersect. A line of centers v_(c)of the pinion axis and the gear axis exists, and a distance (offset)between the two axes on the line of centers v_(c) is set as E, an angle(shaft angle) between the pinion axis and the gear axis projected onto aplane orthogonal to the line of centers v_(c) is set as Σ, and a gearratio is set as i₀. FIG. 1B is a cross sectional diagram at a planeincluding the axis II of the ring gear 14. An angle between a pitch coneelement pc_(g) passing at a design reference point P_(w), to bedescribed later, and the axis II is shown as a pitch cone angle Γ_(gw).A distance between the design reference point P_(w) and the axis II isshown as a reference circle radius R_(2w). FIG. 1C is a cross sectionaldiagram at a plane including the axis I of the pinion 10. An anglebetween a pitch cone element pc_(p) passing at the design referencepoint P_(w) and the axis I is shown as a pitch cone angle γ_(pw). Inaddition, a distance between the design reference point P_(w) and theaxis I is shown as a reference circle radius R_(1w).

FIG. 2 shows coordinate systems C₁ and C₂. A direction of the line ofcenters v_(c) is set to a direction in which the direction of an outerproduct ω_(2i)×ω_(1i) of angular velocities ω_(1i) and ω_(2i) of thepinion gear axis I and the ring gear axis II is positive. Theintersection points of the pinion and ring gear axes I, II and the lineof centers v_(c) are designated by C₁ and C₂ and a situation where C₂ isabove C₁ with respect to the line of centers v_(c) will be considered inthe following. A case where C₂ is below C₁ would be very similar. Adistance between C₁ and C₂ is the offset E. A coordinate system C₂ of aring gear 14 is defined in the following manner. The origin of thecoordinate system C₂ (u_(2c), v_(2c), z_(2c)) is set at C₂, a z_(2c)axis of the coordinate system C₂ is set to extend in the ω₂₀ directionon the ring gear axis II, a v_(2c) axis of the coordinate system C₂ isset in the same direction as that of the line of centers v_(c), and au_(2c) axis of the coordinate system C₂ is set to be normal to both theaxes to form a right-handed coordinate system. A coordinate system C₁(u_(1c), v_(1c), z_(1c)) can be defined in a very similar manner for thepinion 10.

FIG. 3 shows a relationship between the coordinate systems C₁, C₂,C_(q1), and C_(q2) in gears I and II. The coordinate systems O₂ andC_(q2) of the gear II are defined in the following manner. The origin ofthe coordinate system C₂ (u_(2c), v_(2c), z_(2c)) is set at C₂, a z_(2c)axis of the coordinate system C₂ is set to extend in the ψ₂₀ directionon the ring gear axis II, a v_(2c) axis of the coordinate system C₂ isset in the same direction as that of the line of centers v_(c), and au_(2c) axis of the coordinate system C₂ is set to be normal to both theaxes to form a right-handed coordinate system. The coordinate systemC_(q2) (q_(2c), v_(q2c), z_(2c)) has the origin C₂ and the z_(2c) axisin common, and is a coordinate system formed by the rotation of thecoordinate system C₂ around the z_(2c) axis as a rotational axis by χ₂₀(the direction shown in the figure is positive) such that the planev_(2c)=0 is parallel to the plane of action G₂₀. The u_(2c) axis becomesa q_(2c) axis, and the v_(2c) axis becomes a v_(q2c) axis.

The plane of action G₂₀ is expressed by v_(q2c)=−R_(b2) using thecoordinate system C_(q2). In the coordinate system C₂, the inclinationangle of the plane of action G₂₀ to the plane v_(2c)=0 is the angle χ₂₀,and the plane of action G₂₀ is a plane tangent to the base cylinder(radius R_(b20)).

The relationships between the coordinate systems C₂ and C_(q2) become asfollows because the z_(2c) axis is common.

u _(2c) =q _(2c) cos χ₂₀ −v _(q2) sin χ₂₀

v _(2c) =q _(2c) sin χ₂₀ +v _(q2c) cos χ₂₀

Because the plane of action G₂₀ meets v_(q2c)=−R_(b20), the followingexpressions (1), are satisfied if the plane of action G₂₀ is expressedby the radius R_(b20) of the base cylinder.

u _(2c) =q _(2c) cos χ₂₀ +R _(b20) sin χ₂₀

v _(2c) =q _(2c) sin χ₂₀ −R ^(b20) cos χ₂₀

z _(2c) =z _(2c)  (1)

If the line of centers g₀ is defined to be on the plane of action G₂₀and also defined such that the line of centers g₀ is directed in thedirection in which the q_(2c) axis component is positive, an inclinationangle of the line of centers g₀ from the q_(2c) axis can be expressed byψ_(b20) (the direction shown in the figure is positive). Accordingly,the inclination angle of the line of centers g₀ in the coordinatesystem. C₂ is defined to be expressed in the form of g₀ (φ₂₀, ψ_(b20))with the inclination angle φ₂₀ (the complementary angle of the χ₂₀) ofthe plane of action G₂₀ with respect to the line of centers v_(c), andψ_(b2).

As for the gear I, coordinate systems C₁ (u_(1c), v_(1c), z_(1c)) andC_(q1) (g_(1c), v_(q1c), z_(1c)), a plane of action G₁₀, a radius R_(b1)of the base cylinder, and the inclination angle g₀ (φ₁₀, ψ_(b10)) of theline of centers g₀ can be similarly defined. Because the systems share acommon z_(1c) axis, the relationship between the coordinate systems C₁and C_(q1) can also be expressed by the following expressions (2).

u _(1c) =q _(1c) cos χ₁₀ +R _(b10) sin χ₁₀

v _(1c) =q _(1c) sin χ₁₀ −R _(b10) cos χ₁₀

z _(1c) =z _(1c)  (2)

The relationship between the coordinate systems C₁ and C₂ is expressedby the following expressions (3).

u _(1c) =−u _(2c) cos Σ−z _(2c) sin Σ

v _(1c) =v _(2c) +E

z _(1c) =u _(2c) sin Σ−z _(2c) cos Σ  (3)

1.2 Instantaneous Axis (Relative Rotational Axis) S

FIG. 4 shows a relationship between an instantaneous axis and acoordinate system C_(S). If the orthogonal projections of the two axes I(ω₁₀) and II (ω₂₀) to the plane S_(H) are designated by I_(s) (ω₁₀″) andII_(s) (ω₂₀″), respectively, and an angle of I_(S) with respect toII_(S) when the plane S_(H) is viewed from the positive direction of theline of centers v_(c) to the negative direction thereof is designated byΩ, I_(s) is in a zone of 0≦Ω≦π (the positive direction of the angle Ω isthe counterclockwise direction) with respect to II_(S) in accordancewith the definition of ω₂₀×ω₁₀. If an angle of the instantaneous axis S(ω_(r)) to the II_(s) on the plane S_(H) is designated by Ω_(S) (thepositive direction of the angle Ω_(S) is the counterclockwisedirection), the components of ω₁₀″ and ω₂₀″ that are orthogonal to theinstantaneous axis on the plane S_(H) must be equal to each other inaccordance with the definition of the instantaneous axis(ω_(r)=ω₁₀−ω₂₀). Consequently, O_(s) satisfies the following expressions(4):

sin Ω_(s)/sin(Ω_(s)−Ω)=ω₁₀/ω₂₀; or

sin Γ_(s)/sin)Σ−Γ_(s))=ω₁₀/ω₂₀  (4)

wherein Σ=π−Ω (shaft angle) and Γ_(s)=π−Ω_(s). The positive directionsare shown in the figure. In other words, the angle Γ_(s) is aninclination of the instantaneous axis S with respect to the ring gearaxis II_(s) on the plane S_(H), and the angle Γ_(s) will hereinafter bereferred to as an inclination angle of the instantaneous axis.

The location of C_(s) on the line of centers v_(c) can be obtained asfollows. FIG. 5 shows a relative velocity V_(s) (vector) of the pointC_(s). In accordance with the aforesaid supposition, C₁ is located underthe position of C₂ with respect to the line of centers v_(c) andω₁₀≧ω₂₀. Consequently, C_(S) is located under C₂. If the peripheralvelocities of the gears I, II at the point C_(s) are designated byV_(s1) and V_(s2) (both being vectors), respectively, because therelative velocity V_(s) (=V_(s1)−V_(s2)) exists on the instantaneousaxis S, the components of V_(s1) and V_(s2) (existing on the planeS_(H)) orthogonal to the instantaneous axis must always be equal to eachother. Consequently, the relative velocity V_(s) (=V_(s1)−V_(s2)) at thepoint C_(s) would have the shapes as shown in the same figure on theplane S_(H) according to the location (Γ_(s)) of the instantaneous axisS, and the distance C₂C₅ between C₂ and C_(s) can be obtained by thefollowing expression (5). That is,

C ₂ C _(s) =E tan Γ _(s)/{tan(Σ−Γ_(s))+tan Γ_(s)}  (5).

The expression is effective within a range of 0≦Γ_(s)≦π, and thelocation of C_(s) changes together with Γ_(s), and the location of thepoint C_(s) is located above C₁ in the case of 0≦Γ_(s)≦π/2, and thelocation of the point C_(s) is located under C₁ in the case ofπ/2≦Γ_(s)≦π.

1.3 Coordinate System C_(s)

Because the instantaneous axis S can be determined in a static space inaccordance with the aforesaid expressions (4) and (5), the coordinatesystem C_(S) is defined as shown in FIG. 4. The coordinate system C_(S)(u_(c), v_(c), z_(c)) is composed of C_(s) as its origin, the directedline of centers v_(c) as its v_(c) axis, the instantaneous axis S as itsz_(c) axis (the positive direction thereof is the direction of ω_(r)),and its u_(c) axis taken to be normal to both the axes as a right-handedcoordinate system. Because it is assumed that a pair of gears beingobjects transmit a motion of a constant ratio of angular velocity, thecoordinate system C_(s) becomes a coordinate system fixed in the staticspace, and the coordinate system C_(s) is a basic coordinate system inthe case of treating a pair of gears performing the transmission of themotion of constant ratio of angular velocity together with thepreviously defined coordinate systems C₁ and C₂.

1.4 Relationship Among Coordinate Systems C₁, C₂, and C_(s)

If the points C₁ and C₂ are expressed to be C₁ (0, v_(cs1), 0) and C₂(0, v_(cs2), 0) by the use of the coordinate system C_(s), v_(cs1) andv_(cs2) are expressed by the following expressions (6).

$\begin{matrix}{{v_{{cs}\; 2} = {{C_{S}C_{2}} = {E\; \tan \; {\Gamma_{s}/\left\{ {{\tan \left( {\Sigma - \Gamma_{s}} \right)} + {\tan \; \Gamma_{s}}} \right\}}}}}\begin{matrix}{v_{{cs}\; 1} = {C_{S}C_{1}}} \\{= {v_{{cs}\; 2} - E}} \\{= {{- E}\; {{\tan \left( {\Sigma - \Gamma_{s}} \right)}/\left\{ {{\tan \left( {\Sigma - \Gamma_{s}} \right)} + {\tan \; \Gamma_{s}}} \right\}}}}\end{matrix}} & (6)\end{matrix}$

If it is noted that C₂ is always located above C_(s) with respect to thev_(c) axis, the relationships among the coordinate system C_(s) and thecoordinate systems C₁ and C₂ can be expressed as the followingexpressions (7) and (8) with the use of v_(cs1), v_(cs2), Σ, and Γ_(s).

u _(1c) =u _(c) cos(Σ−Γ_(s))+z _(c) sin(Σ−Γ_(s))

v _(1c) =v _(c) −v _(cs1)

z _(1c) =−u _(c) sin(Σ−Γ_(s))+z _(c) cos(Σ−Γ_(s))  (7)

u _(2c) =−u _(c) cos Γ_(s) +z _(c) sin Γ_(s)

v _(2c) =v _(c) −v _(cs2)

z _(2c) =−u _(c) sin Γ_(s) −z _(c) cos Γ_(s)  (8)

The relationships among the coordinate system C_(s) and the coordinatesystems C₁ and C₂ are conceptually shown in FIG. 6.

2. Definition of Path of Contact G₀ by Coordinate System C_(s)

2.1 Relationship Between Relative Velocity and Path of Contact g₀

FIG. 7 shows a relationship between the set path of contact g₀ and arelative velocity V_(rs) (vector) at an arbitrary point P on g₀.Incidentally, a prime sign (′) and a double-prime sign (″) in the figureindicate orthogonal projections of a point and a vector on the targetplane. If the position vector of the P from an arbitrary point on theinstantaneous axis S is designated by r when a tooth surface contacts atthe arbitrary point P on the path of contact g₀, the relative velocityV_(rs) at the point P can be expressed by the following expression (9).

v _(rs)=ω_(r) ×r+V _(s)  (9)

where

ω_(r)=ω₁₀−ω₂₀

ω_(r)=ω₂₀ sin Σ/sin(Σ−Γ_(s))=ω₁₀ sin Σ/sin Γ_(s)

V_(s)=ω₁₀×[C₁S_(s)]−ω₂₀×[C₂C_(s)]

V_(s)=ω₂₀E sin Γ_(s)=ω₁₀E sin(Σ−Γ_(s)).

Here, [C₁C_(s)] indicates a vector having C₁ as its starting point andC_(s) as its endpoint, and [C₂C_(s)] indicates a vector having C₂ as itsstarting point and C_(s) as its end point.

The relative velocity V_(rs) exists on a tangential plane of the surfaceof a cylinder having the instantaneous axis S as an axis, and aninclination angle ψ relative to V_(s) on the tangential plane can beexpressed by the following expression (10).

cos ψ=|V_(s) |/|V _(rs)|  (10)

Because the path of contact g₀ is also the line of centers of a toothsurface at the point of contact, g₀ is orthogonal to the relativevelocity V_(rs) at the point P. That is,

V _(rs) ·g ₀=0

Consequently, g₀ is a directed straight line on a plane N normal toV_(rs) at the point P. If the line of intersection of the plane N andthe plane S_(H) is designated by H_(n), H_(n) is in general a straightline intersecting with the instantaneous axis S, with g₀ necessarilypassing through the H_(n) if an infinite intersection point is included.If the intersection point of g₀ with the plane S_(H) is designated byP₀, then P₀ is located on the line of intersection H_(n), and g₀ and P₀become as follows according to the kinds of pairs of gears.

(1) Case of Cylindrical Gears or Bevel Gears (Σ=0, π or E=0)

Because V_(s)=0, V_(rs) simply means a peripheral velocity around theinstantaneous axis S. Consequently, the plane N includes the S axis.Hence, H_(n) coincides with S, and the path of contact g₀ always passesthrough the instantaneous axis S. That is, the point P₀ is located onthe instantaneous axis S. Consequently, for these pairs of gears, thepath of contact g₀ is an arbitrary directed straight line passing at thearbitrary point P₀ on the instantaneous axis.

(2) Case of Gear Other than that Described Above (Σ≠0, π or E≠0)

In the case of a hypoid gear, a crossed helical gear or a wormgear, ifthe point of contact P is selected at a certain position, the relativevelocity V_(rs), the plane N, and the straight line H_(n), all peculiarto the point P, are determined. The path of contact g₀ is a straightline passing at the arbitrary point P₀ on H_(n), and does not, ingeneral, pass through the instantaneous axis S. Because the point P isarbitrary, g₀ is also an arbitrary directed straight line passing at thepoint P₀ on a plane normal to the relative velocity V_(rs0) at theintersection point P₀ with the plane S_(H). That is, the aforesaidexpression (9) can be expressed as follows.

V _(rs) =V _(rs0)+ω_(r) ×[P ₀ P]·g ₀

Here, [P₀P] indicates a vector having P₀ as its starting point and the Pas its end point. Consequently, if V_(rs0)·g₀=0, V_(rs)·g₀=0, and thearbitrary point P on g₀ is a point of contact.

2.2 Selection of Reference Point

Among pairs of gears having two axes with known positional relationshipand the angular velocities, pairs of gears with an identical path ofcontact g₀ have an identical tooth profile corresponding to g₀, with theonly difference between them being which part of the tooth profile isused effectively. Consequently, in design of a pair of gears, theposition at which the path of contact g₀ is disposed in a static spacedetermined by the two axes is important. Further, because a designreference point is only a point for defining the path of contact g₀ inthe static space, the position at which the design reference point isselected on the path of contact g₀ does not cause any essentialdifference. When an arbitrary path of contact g₀ is set, the g₀necessarily intersects with a plane S_(H) including the case where theintersection point is located at an infinite point. Thus, the path ofcontact g₀ is determined with the point P₀ on the plane S_(H) (on aninstantaneous axis in the case of cylindrical gears and bevel gears) asthe reference point.

FIG. 8 shows the reference point P₀ and the path of contact g₀ by theuse of the coordinate system C_(s). When the reference point expressedby means of the coordinate system C_(s) is designated by P₀ (u_(c0),v_(c0), z_(c0)), each coordinate value can be expressed as follows.

u _(c0) =O _(s) P ₀

v _(c0)=0

z _(c0) =C _(s) O _(s)

For cylindrical gears and bevel gears, u_(c0)=0. Furthermore, the pointO_(s) is the intersection point of a plane S_(s), passing at thereference point P₀ and being normal to the instantaneous axis S, and theinstantaneous axis S.

2.3 Definition of Inclination Angle of Path of Contact g₀

The relative velocity V_(rs0) at the point P₀ is concluded as followswith the use of the aforesaid expression (9).

V _(rs0)=ω_(r) ×[u _(c0) ]+V _(s)

where, [u_(c0)] indicates a vector having O_(s) as its starting pointand P₀ as its end point. If a plane (u_(c)=u_(c0)) being parallel to theinstantaneous axis S and being normal to the plane S_(H) at the point P₀is designated by S_(p), V_(rs0) is located on the plane S_(p), and theinclination angle ψ₀ of V_(rs0) from the plane S_(H) (v_(c)=0) can beexpressed by the following expression (11) with the use of the aforesaidexpression (10).

$\begin{matrix}\begin{matrix}{{\tan \; \psi_{0}} = {\omega_{r}{u_{c\; 0}/V_{s}}}} \\{= {u_{c\; 0}\sin \; {\Sigma/\left\{ {E\; {\sin \left( {\Sigma - \Gamma_{s}} \right)}\sin \; \Gamma_{s}} \right\}}}}\end{matrix} & (11)\end{matrix}$

Incidentally, ψ₀ is assumed to be positive when u_(c0)≧0, and thedirection thereof is shown in FIG. 8.

If a plane passing at the point P₀ and being normal to V_(rs0) isdesignated by S, the plane S_(n) is a plane inclining to the plane S_(s)by the ψ₀, and the path of contact g₀ is an arbitrary directed straightline passing at the point P₀ and located on the plane S_(n).Consequently, the inclination angle of g₀ in the coordinate system C_(s)can be defined with the inclination angle ψ₀ of the plane S_(n) from theplane S_(s) (or the v_(c) axis) and the inclination angle  _(n0) fromthe plane S_(p) on the plane S_(n), and the defined inclination angle isdesignated by g₀ (ψ₀, φ_(n0)). The positive direction of φ_(n0) is thedirection shown in FIG. 8.

2.4. Definition of g₀ by Coordinate System C_(s)

FIG. 6 shows relationships among the coordinate system C_(s), the planesS_(H), S_(s), S_(p) and S_(n), P₀ and g₀ (ψ₀, φn₀). The plane S_(H)defined here corresponds to a pitch plane in the case of cylindricalgears and an axial plane in the case of a bevel gear according to thecurrent theory. The plane S_(s) is a transverse plane, and the planeS_(p) corresponds to the axial plane of the cylindrical gears and thepitch plane of the bevel gear. Furthermore, it can be considered thatthe plane S_(n) is a normal plane expanded to a general gear, and thatφ_(n0) and ψ₀ also are a normal pressure angle and a spiral angleexpanded to a general gear, respectively. By means of these planes,pressure angles and spiral angles of a pair of general gears can beexpressed uniformly to static spaces as inclination angles to each planeof line of centers (g₀'s in this case) of points of contact. The planesS_(n), φ_(n0), and ψ₀ defined here coincide with those of a bevel gearof the current theory, and differ for other gears because the currenttheory takes pitch planes of individual gears as standards, and then thestandards change to a static space according to the kinds of gears. Withthe current theory, if a pitch body of revolution (a cylinder or acircular cone) is determined, it is sufficient to generate a matingsurface by fixing an arbitrary curved surface to the pitch body ofrevolution as a tooth surface, and in the current theory, conditions ofthe tooth surface (a path of contact and the normal thereof) are notlimited except for the limitations of manufacturing. Consequently, thecurrent theory emphasizes the selection of P₀ (for discussions aboutpitch body of revolution), and there has been little discussionconcerning design of g₀ (i.e. a tooth surface realizing the g₀) beyondthe existence of a tooth surface.

For a pair of gears having the set shaft angle Σ thereof, the offset Ethereof, and the directions of angular velocities, the path of contactg₀ can generally be defined in the coordinate system C_(s) by means offive independent variables of the design reference point P₀ (u_(c0),v_(c0), z_(c0)) and the inclination angle g₀ (ψ₀, φ_(n0)). Because theratio of angular velocity i₀ and v_(c0)=0 are set as design conditionsin the present embodiment, there are three independent variables of thepath of contact g₀. That is, the path of contact g₀ is determined in astatic space by the selections of the independent variables of two of(z_(c0)), φ_(n0), and ψ₀ in the case of cylindrical gears because z_(c0)has no substantial meaning, three of z_(c0), φ_(n0), and ψ₀ in the caseof a bevel gear, or three of z_(c0), φ_(n0), and ψ₀ (or u_(c0)) in thecase of a hypoid gear, a worm gear, or a crossed helical gear. When thepoint P₀ is set, ψ₀ is determined at the same time and only φ_(n0) is afreely selectable variable in the case of the hypoid gear and the wormgear. However, in the case of the cylindrical gears and the bevel gear,because P₀ is selected on an instantaneous axis, both of ψ₀ and φ_(n0)are freely selectable variables.

3. Pitch Hyperboloid 3.1 Tangential Cylinder of Relative Velocity

FIG. 9 is a diagram showing an arbitrary point of contact P_(w), acontact normal g_(w) thereof, a pitch plane S_(tw), the relativevelocity V_(rsw), and a plane S_(nw) which is normal to the relativevelocity V_(rsw) of a hypoid gear, along with basic coordinate systemsC₁, C₂, and C_(s). FIG. 10 is a diagram showing FIG. 9 drawn from apositive direction of the z_(c) axis of the coordinate system C_(s). Thearbitrary point P_(w) and the relative velocity V_(rsw) are shown withcylindrical coordinates P_(w)(r_(w), β_(w), z_(cw): C_(s)). The relativevelocity V_(rsw) is inclined by ψ_(rw) from a generating line L_(pw) onthe tangential plane S_(pw) of the cylinder having the z, axis as itsaxis, passing through the arbitrary point P_(w), and having a radius ofr_(w).

When the coordinate system C_(s) is rotated around the z_(c) axis byβ_(w), to realize a coordinate system C_(rs) (u_(rc), v_(rc), z_(c):C_(rs)), the tangential plane S_(pw) can be expressed by u_(rc)=r_(w),and the following relationship is satisfied between u_(rc)=r_(w) and theinclination angle ω_(rw) of V_(rsw).

$\begin{matrix}\begin{matrix}{u_{rc} = r_{w}} \\{= {V_{s}\tan \; {\psi_{rw}/\omega_{r}}}} \\{= {E\; \tan \; \psi_{rw} \times {\sin \left( {\Sigma - \Gamma_{s}} \right)}\sin \; {\Gamma_{s}/\sin}\; \Sigma}}\end{matrix} & (12)\end{matrix}$

where Vs represents a sliding velocity in the direction of theinstantaneous axis and ω_(r) represents a relative angular velocityaround the instantaneous axis.

The expression (12) shows a relationship between r_(w) of the arbitrarypoint P_(w) (r_(w), β_(w), z_(cw): C_(s)) and the inclination angleψ_(rw) of the relative velocity V_(rsw) thereof. In other words, whenψ_(rw) is set, r_(w) is determined. Because this is true for arbitraryvalues of β_(w) and z_(cw), P_(w) with a constant ψ_(rw) defines acylinder with a radius r_(w). This cylinder is called the tangentialcylinder of the relative velocity.

3.2 Pitch Generating Line and Surface of Action

When r_(w) (or ψ_(rw)) and β_(w) are set, P_(w) is determined on theplane z_(c)=z_(cw). Because this is true for an arbitrary value ofz_(cw), points P_(w) having the same r_(w) (or ψ_(rw)) and the sameβ_(w) draw a line element of the cylinder having a radius r_(w). Thisline element is called a pitch generating line L_(pw). A directedstraight line which passes through a point P_(w) on a plane S_(nw)orthogonal to the relative velocity V_(rsw) at the arbitrary point P_(w)on the pitch generating line L_(pw) satisfies a condition of contact,and thus becomes a contact normal.

FIG. 11 is a diagram conceptually drawing relationships among the pitchgenerating line L_(pw), directed straight line g_(w), surface of actionS_(w), contact line w, and a surface of action S_(wc) (dotted line) onthe side C to be coast. A plane having an arbitrary directed straightline g_(w) on the plane S_(nw) passing through the point P_(w) as anormal is set as a tooth surface W. Because all of the relative velocityV_(rsw) at the arbitrary point P_(w) on the pitch generating line L_(pw)are parallel and the orthogonal planes S_(nw) are also parallel, of thenormals of the tooth surface W, any normal passing through the pitchgenerating line L_(pw) becomes a contact normal, and a plane defined bythe pitch generating line L_(pw) and the contact normal g_(w) becomesthe surface of action S_(w) and an orthogonal projection of the pitchgenerating line L_(pw) to the tooth surface W becomes the contact linew. Moreover, because the relationship is similarly true for anothernormal g_(wc) on the plane S_(w) passing through the point P_(w) and thesurface of action S_(wc) thereof, the pitch generating line L_(pw) is aline of intersection between the surfaces of action of two toothsurfaces (on the drive-side D and coast-side C) having different contactnormals on the plane S_(nw).

3.3 Pitch Hyperboloid

The pitch generating line L_(pw) is uniquely determined by the shaftangle Σ, offset E, gear ratio i₀, inclination angle ψ_(rw) of relativevelocity V_(rsw), and rotation angle β_(w) from the coordinate systemC_(s) to the coordinate system C_(rs). A pair of hyperboloids which areobtained by rotating the pitch generating line L_(pw) around the twogear axes, respectively, contact each other in a line along L_(pw), andbecause the line L_(pw) is also a line of intersection between thesurfaces of action, the drive-side D and the coast-side C also contacteach other along the line L_(pw). Therefore, the hyperboloids are suitedas revolution bodies for determining the outer shape of the pair ofgears. In the present invention, the hyperboloids are set as the designreference revolution bodies, and are called the pitch hyperboloids. Thehyperboloids in the related art are revolution bodies in which theinstantaneous axis S is rotated around the two gear axes, respectively,but in the present invention, the pitch hyperboloid is a revolution bodyobtained by rotating a parallel line having a distance r_(w) from theinstantaneous axis.

In the cylindrical gear and the bevel gear, L_(pw) coincides with theinstantaneous axis S or z_(c) (r_(w)→0) regardless of ψ_(rw) and β_(w),because of special cases of the pitch generating line L_(pw) (V_(s)→0 asΣ→0 or E→0 in the expression (12)). The instantaneous axis S is a lineof intersection of the surfaces of action of the cylindrical gear andthe bevel gear, and the revolution bodies around the gear axes are thepitch cylinder of the cylindrical gear and the pitch cone of the bevelgear.

For these reasons, the pitch hyperboloids which are the revolutionbodies of the pitch generating line L_(pw) have the common definition ofthe expression (12) from the viewpoint that the hyperboloid is a“revolution body of line of intersection of surfaces of action” and canbe considered to be a design reference revolution body for determiningthe outer shape of the pair of gears which are common to all pairs ofgears.

3.4 Tooth Trace (New Definition of Tooth Trace)

In the present invention, a curve on the pitch hyperboloid (which iscommon to all gears) obtained by transforming a path of contact to acoordinate system which rotates with the gear when the tooth surfacearound the point of contact is approximated with its tangential planeand the path of contact is made coincident with the line of intersectionof the surfaces of action (pitch generating line L_(pw)) is called atooth trace (curve). In other words, a. tooth profile, among arbitrarytooth profiles on the tooth surface, in which the path of contactcoincides with the line of intersection of the surface of action iscalled a tooth trace. The tooth trace of this new definition coincideswith the tooth trace of the related art defined as an intersectionbetween the pitch surface (cone or cylinder) and the tooth surface inthe cylindrical gears and the bevel gears and differs in other gears. Inthe case of the current hypoid gear, the line of intersection betweenthe selected pitch cone and the tooth surface is called a tooth trace.

3.5 Contact Ratio

A total contact ratio m is defined as a ratio of a maximum angulardisplacement and an angular pitch of a contact line which moves on aneffective surface of action (or effective tooth surface) with therotation of the pair of gears. The total contact ratio m can beexpressed as follows in terms of the angular displacement of the gear.

m=(θ_(2max)−θ_(2min))/(2θ_(2p))

where θ_(2max) and θ_(2min) represent maximum and minimum gear angulardisplacements of the contact line and 2θ_(2p) represents a gear angularpitch.

Because it is very difficult to represent the position of the contactline as a function of a rotation angle except for special cases(involute helicoid) and it is also difficult to represent such on thetooth surface (curved surface), in the stage of design, the surface ofaction has been approximated with a plane in a static space, a path ofcontact has been set on the surface of action, and an contact ratio hasbeen determined and set as an index along the path of contact.

FIGS. 12 and 13 show the surface of action conceptually shown in FIG. 11in more detail with reference to the coordinate systems C_(s), C₁, andC₂. FIG. 12 shows a surface of action in the cases of the cylindricalgear and a crossed helical gear, and FIG. 13 shows a surface of actionin the cases of the bevel gear and the hypoid gear. FIGS. 12 and 13 showsurfaces of action with the tooth surface (tangential plane) when theintersection between g, and the reference plane S_(H) (v_(c)=0) is setas P₀ (u_(c0), v_(c0)=0, z_(c0): C_(s)), the inclination angle of g_(w)is represented in the coordinate system C_(s), and the contact normal g,is set g_(w)=g₀ (ψ₀, φ_(n0): C_(s)). A tooth surface passing through P₀is shown as W₀, a tooth surface passing through an arbitrary point P_(d)on g_(w)=g₀ is shown as W_(d), a surface of action is shown byS_(w)=S_(w0), and an intersection between the surface of action and theplane S_(H) is shown with L_(pw0) (which is parallel to L_(pw)). Becauseplanes are considered as the surface of action and the tooth surface,the tooth surface translates on the surface of action. The point P_(w)may be set at any point, but because the static coordinate system C_(s)has its reference at the point P₀ on the plane S_(H), the contact ratiois defined with an example configuration in which P_(w) is set at P₀.

The contact ratio of the tooth surface is defined in the followingmanner depending on how the path of contact passing through P_(w)=P₀ isdefined on the surface of action S_(w)=S_(w0):

(1) Contact Ratio m_(z) Orthogonal Axis

This is a ratio between a length separated by an effective surface ofaction (action limit and the tooth surface boundary) of lines ofintersection h_(1z) and h_(2z) (P₀P_(z1sw) and P₀P_(z2sw) in FIGS. 12and 13) between the surface of action S_(w0) and the planes of rotationZ₁₀ and Z₂₀ and a pitch in this direction;

(2) Tooth Trace Contact Ratio m_(f)

This is a ratio between a length of L_(pw0) which is parallel to theinstantaneous axis separated by the effective surface of action and apitch in this direction;

(3) Transverse Contact Ratio m_(s)

This is a ratio between a length separated by an effective surface ofaction of a line of intersection (P₀P_(ssw) in FIGS. 12 and 13) betweena plane S_(s) passing through P₀ and normal to the instantaneous axisand S_(w0) and a pitch in this direction;

(4) Contact Ratio in Arbitrary Direction

This includes cases where the path of contact is set in a direction ofg₀ (P₀P_(Gswn) in FIG. 13) and cases where the path of contact is set ina direction of a line of intersection (P_(w)P_(gcon) in FIGS. 12 and 13)between an arbitrary conical surface and S_(w0);

(5) Total Contact Ratio

This is a sum of contact ratios in two directions (for example, (2) and(3)) which are normal to each other on the surface of action, and isused as a substitute for the total contact ratio.

In addition, except for points on g_(w)=g₀, the pitch (length) woulddiffer depending on the position of the point, and the surface of actionand the tooth surface are actually not planes. Therefore, only anapproximated value can be calculated for the contact ratio. Ultimately,a total contact ratio determined from the angular displacement must bechecked.

3.6 General Design Method of Gear Using Pitch Hyperboloid

In general, a gear design can be considered, in a simple sense, to be anoperation, in a static space (coordinate system C_(s)) determined bysetting the shaft angle Σ, offset E, and gear ratio i₀, to:

(1) select a pitch generating line and a design reference revolutionbody (pitch hyperboloid) by setting a design reference point P_(w)(r_(w)(ψ_(rw)), β_(w), z_(cw): C_(rs)); and(2) set a surface of action (tooth surface) having g_(w) by setting aninclination angle (ψ_(rw), φ_(nrw): C_(rs)) of a tooth surface normalg_(w) passing through P_(w).

In other words, the gear design method (selection of P_(w) and g_(w))comes down to selection of four variables including r_(w) (normally,ψ_(rw) is set), β_(w), z_(cw) (normally, R_(2w) (gear pitch circleradius) is set in place of z_(cw)), and φ_(nrw). A design method for ahypoid gear based on the pitch hyperboloid when Σ, E, and i₀ are setwill be described below.

3.7 Hypoid Gear (−π/2<β_(w)<π/2)

(1) Various hypoid gears can be realized depending on how β_(w) isselected, even with set values for ψ_(rw) (r_(w)) and z_(cw) (R_(2w)).

(a) From the viewpoint of the present invention, the Wildhaber (Gleason)method is one method of determining P_(w) by determining β_(w) throughsetting of a constraint condition to “make the radius of curvature of atooth trace on a plane (FIG. 9) defined by peripheral velocities of apinion and ring gear at P_(w) coincide with the cutter radius”. However,because the tooth surface is possible as long as an arbitrary curvedsurface (therefore, arbitrary radius of curvature of tooth trace) havingg_(w) passing through P_(w) has a mating tooth surface, this conditionis not necessarily a requirement even when a conical cutter is used. Inaddition, although this method employs circular cones which circumscribeat P_(w), the pair of gears still contact on a surface of action havingthe pitch generating line L_(pw) passing through P_(w) regardless of thecones. Therefore, the line of intersection between the pitch conecircumscribing at P_(w) and the surface of action determined by thismethod differs from the pitch generating line L_(pw) (line ofintersection of surfaces of action). When g, is the same, theinclination angle of the contact line on the surface of action andL_(pw) are equal to each other, and thus the pitch in the direction ofthe line of intersection between the surface of action and the pitchcone changes according to the selected pitch cone (FIG. 11). In otherwords, a large difference in the pitch is caused between the drive-sideand the coast-side in the direction of the line of intersection betweenthe pitch cone and the surface of action (and, consequently, the contactratios in this direction). In the actual Wildhaber (Gleason) method, twocones are determined by giving pinion spiral angle and an equation ofradius of curvature of tooth trace for contact equations of the twocones (seven equations having nine unknown variables), and thus theexistence of the pitch generating line and the pitch hyperboloid is notconsidered.

(b) In a preferred embodiment described in section 4.2A below, β_(w) isselected by giving a constraint condition that “a line of intersectionbetween a cone circumscribing at P_(w) and the surface of action iscoincident with the pitch generating line L_(pw)”. As a result, as willbe described below, the tooth trace contact ratios on the drive-side andthe coast-side become approximately equal to each other.

(2) Gear radius R_(2w), β_(w), and ψ_(rw) are set and a design referencepoint P_(w) (u_(cw), v_(cw), z_(cw): C_(s)) is determined on the pitchgenerating line L_(pw). The pitch hyperboloids can be determined byrotating the pitch generating line L_(pw) around each tooth axis. Amethod of determining the design reference point will be described insection 4.2B below.

(3) A tooth surface normal g_(w) passing through P_(w) is set on a planeS_(nw) normal to the relative velocity V_(rsw) of P_(w). The surface ofaction S_(w) is determined by g_(w) and the pitch generating lineL_(pw).

4. Design Method for Hypoid Gear

A method of designing a hypoid gear using the pitch hyperboloid will nowbe described in detail.

4.1 Coordinate Systems C_(s), C₁, and C₂ and Reference Point P_(w)

When the shaft angle Σ, offset E, and gear ratio i₀ are set, theinclination angle Γ_(s) of the instantaneous axis, and the origins C₁(0, v_(cs1), 0: C_(s)) and C₂ (0, v_(cs2), 0: C_(s)) of the coordinatesystems C₁ and C₂ are represented by the following expressions.

sin Γ_(s)/sin(Σ−Γ_(s))=i ₀

v _(cs2) =E tan Γ _(s)/{tan(Σ−Γ_(s))+tan Γ_(s)}

v _(cs1) =v _(cs2) −E

The reference point P_(w) is set in the coordinate system C_(s) asfollows.

P _(w)(u _(cw) ,v _(cw) ,z _(cw) :C _(s))

If P_(w) is set as P_(w)(r_(w), β_(w), z_(cw): C_(s)) by representingP_(w) with the cylindrical radius r_(w) of the relative velocity and theangle from the u_(c) axis, the following expressions hold.

u _(cw) =r _(w) cos β_(w)

v _(cw) =r _(w) sin β_(w)

The pitch generating line L_(pw) is determined as a straight line whichpasses through the reference point P_(w) and which is parallel to theinstantaneous axis (inclination angle Γ_(s)), and the pitch hyperboloidsare determined as revolution bodies of the pitch generating line L_(pw)around the gear axes.

If the relative velocity of P_(w) is V_(rsw), the angle ψ_(rw) betweenV_(rsw) and the pitch generating line L_(pw) is, based on expression(12),

tan ψ_(rw) =r _(w) sin Σ/{E sin(Σ−Γ_(s))sin Γ_(s)}

Here, ψ_(rw) is the same anywhere on the same cylinder of the radiusr_(w).

When transformed into coordinate systems C₁ and C₂, P_(w) (u_(1cw),v_(1cw), z_(1cw): C₁), P_(w)(u_(2cw), v_(2cw), z_(2cw): C₂), and pinionand ring gear reference circle radii R_(1w) and R_(2w) can be expressedwith the following expressions.

u _(1cw) =u _(cw) cos(Σ−Γ_(s))+z _(cw) sin(Σ−Γ_(s))

v _(1cw) =v _(cw) −v _(cs1)

z _(1cw) =−u _(cw) sin(Σ−Γ_(s))+z _(cw) cos(Σ−Γ_(s))

u _(2ce) =−u _(cw) cos Γ_(s) +z _(cw) sin Γ_(s)

v _(2cw) =v _(cw) −v _(cs2)

z _(2cw) =−u _(cw) sin Γ_(s) −z _(cw) cos Γ_(s)

R _(1w) ² =u _(1cw) ² +v _(1cw) ²

R _(2w) ² =u _(2cw) ² +v _(2cw) ²  (13)

4.2A Cones Passing Through Reference Point P_(w)

A pitch hyperboloid which is a geometric design reference revolutionbody is difficult to manufacture, and thus in reality, in general, thegear is designed and manufactured by replacing the pitch hyperboloidwith a pitch cone which passes through the point of contact P_(w). Thereplacement with the pitch cones is realized in the present embodimentby replacing with cones which contact at the point of contact P_(w).

The design reference cone does not need to be in contact at P_(w), butcurrently, this method is generally practiced. When β_(w) is changed,the pitch angle of the cone which contacts at P_(w) changes in variousmanners, and therefore another constraint condition is added forselection of the design reference cone (β_(w)). The design method woulddiffer depending on the selection of the constraint condition. One ofthe constraint conditions is the radius of curvature of the tooth tracein the Wildhaber (Gleason) method which is already described. In thepresent embodiment, β_(w) is selected with a constraint condition that aline of intersection between the cone which contacts at P_(w) and thesurface of action coincides with the pitch generating line L_(pw).

As described, there is no substantial difference caused by where on thepath of contact g₀ the design reference point is selected. Therefore, adesign method of a hypoid gear will be described in which the point ofcontact P_(w) is set as the design reference point and circular coneswhich contact at P_(w) are set as the pitch cones.

4.2A.1 Pitch Cone Angles

Intersection points between a plane S_(nw) normal to the relativevelocity V_(rsw) of the reference point P_(w) and the gear axes are setas O_(1nw) and O_(2nw) (FIG. 9). FIG. 14 is a diagram showing FIG. 9viewed from the positive directions of the tooth axes z_(1c) and z_(2c),and intersections O_(1nw) and O_(2nw) can be expressed by the followingexpressions.

O _(1nw)(0,0,−E/(tan ε_(2w) sin Σ):C ₁)

O _(2nw)(0,0,−E/(tan ε_(1w) sin Σ):C ₂)

where sin ε_(1w)=v_(1cw)/R_(1w) and sin ε_(2w)=v_(2cw)/R_(2w).

In addition, O_(1nw)P_(w) and O_(2nw)P_(w) can be expressed with thefollowing expressions.

O _(1nw) P _(w) ={R _(1w) ²+(−E/(tan ε_(2w) sin Σ)−z _(1cw))²}^(1/2)

O _(2nw) P _(w) ={R _(2w) ²+(−E/(tan ε_(1w) sin Σ)−z _(2cw))²}^(1/2)

Therefore, the cone angles γ_(pw) and Γ_(gw) of the pinion and ring gearcan be determined with the following expressions, taking advantage ofthe fact that O_(1nw)P_(w) and O_(2nw)P_(w) are back cone elements:

cos γ_(pw) =R _(1w) /O _(1nw) P _(w)

cos Γ_(gw) =R _(2w) /O _(2nw) P _(w)  (14)

The expression (14) sets the pitch cone angles of cones having radii ofR_(1w) and R_(2w) and contacting at P_(w).

4.2A.2 Inclination Angle of Relative Velocity at Reference Point P_(w)

The relative velocity and peripheral velocity are as follows.

V _(rsw)/ω₂₀ ={E sin Γ_(s))²+(r _(w) sin Σ/sin(Σ−Γ_(s)))²}^(1/2)

V _(1w)/ω₂₀ =i ₀ R _(1w)

V _(2w)/ω₂₀ =R _(2w)

When a plane defined by peripheral velocities V_(1w) and V_(2w) isS_(tw), the plane S_(tw) is a pitch plane. If an angle formed by V_(1w)and V_(2w) is ψ_(v12w) and an angle formed by V_(rsw) and V_(1w) isψ_(vrs1w) (FIG. 9),

cos(ψ_(v12w))=(V _(1w) ² +V _(2w) ² −V _(rsw) ²)/(2V _(1w) ×V _(2w))

cos(ψ_(vrs1w))=(V _(rsw) ² +V _(1w) ² −V _(2w) ²)/(2V _(1w) ×V _(rsw))

If the intersections between the plane S_(t) and the pinion and gearaxes are O_(1w) and O_(2w), the spiral angles of the pinion and the ringgear can be determined in the following manner as inclination angles onthe plane S_(tw) from P_(w)O_(1w) and P_(w)O_(2w) (FIG. 9).

ψ_(pw)=π/2−ψ_(vrs1w)

ψ_(gw)=π/2−ψ_(v12w)−ψ_(vrs1w)  (15)

When a pitch point P_(w)(r_(w), β_(w), z_(cw): C_(s)) is set,specifications of the cones contacting at P_(w) and the inclinationangle of the relative velocity V_(rsw) can be determined based onexpressions (13), (14) and (15). Therefore, conversely, the pitch pointP_(w) and the relative velocity V_(rsw) can be determined by settingthree variables (for example, R_(2w), ψ_(pw), Γ_(gw)) from among thecone specifications and the inclination angle of the relative velocityV_(rsw). Each of these three variables may be any variable as long asthe variable represents P_(w), and the variables may be, in addition tothose described above, for example, a combination of a ring gearreference radius R_(2w), a ring gear spiral angle ψ_(gw), and a gearpitch cone angle Γ_(gw), or a combination of the pinion reference radiusR_(1w), the ring gear spiral angle ψ_(pw), and Γ_(gw).

4.2A.3 Tip Cone Angle

Normally, an addendum a_(G) and an addendum angleα_(G)=a_(G)/O_(2w)P_(w) are determined and the tip cone angle isdetermined by Γ_(gf)=Γ_(s)+α_(G). Alternatively, another value may bearbitrarily chosen for the addendum angle α_(G).

4.2A.4 Inclination Angle of Normal g_(W) at Reference Point P_(w)

FIG. 15 shows the design reference point P_(w) and the contact normalg_(w) on planes S_(tw), S_(nw), and G_(2w).

(1) Expression of Inclination Angle of g_(W) in Coordinate System C_(s)

An intersection between g_(w) passing through P_(w)(u_(cw), v_(cw),z_(cw): C_(s)) and the plane S_(H) (β_(w)=0) is set as P₀ (u_(c0), 0,z_(c0): C_(s)) and the inclination angle of g_(w) is represented withreference to the point P₀ in the coordinate system C_(s), by g_(w) (ψ₀,φ_(n0): C_(s)). The relationship between P₀ and P_(w) is as follows(FIG. 11):

u _(c0) =u _(cw)+(v _(cw)/cos ψ₀)tan φ_(n0)

z _(c0) =z _(cw) −v _(cw) tan ψ₀  (16)

(2) Expression of Inclination Angle of g_(w) on Pitch Plane S_(tw) andPlane S_(nw) (FIG. 9)

When a line of intersection between the plane S_(nw) and the pitch planeS_(tw) is g_(tw), an inclination angle on the plane S_(nw) from g_(tw)is set as φ_(nw). The inclination angle of g_(w) is represented by g_(w)(ψ_(gw), φ_(nw)) using the inclination angle ψ_(gw) of V_(rsw) fromP_(w)O_(2w) on the pitch plane S_(tw) and φ_(nw).

(3) Transformation Equation of Contact Normal g_(w)

In the following, transformation equations from g_(w) (ψ_(gw), φ_(nw))to g_(w) (ψ₀, φ_(n0): C_(s)) will be determined.

FIG. 15 shows g_(w) (ψ_(gw), φ_(nw)) with g_(w) (φ_(2w), ψ_(b2w): C₂).In FIG. 15, g_(w) is set with P_(w)A, and projections of point A aresequentially shown with B, C, D, and E. In addition, the projectionpoints to the target sections are shown with prime signs (′) anddouble-prime signs (″). The lengths of the directed line segments aredetermined in the following manner, with P_(w)A=L_(g):

$\begin{matrix}{\mspace{79mu} {{{A^{\prime}A} = {L_{g}\sin \; \phi_{nw}}}\mspace{79mu} {{B^{\prime}B} = {L_{g}\cos \; \phi_{nw}\cos \; \psi_{gw}}}\mspace{79mu} {{C^{\prime}C} = {A^{\prime}A}}\mspace{79mu} {{P_{w}C^{\prime}} = {L_{g}\cos \; \phi_{nw}\sin \; \psi_{gw}}}\mspace{79mu} {{C^{\prime}K} = {P_{w}{C^{\prime}/\tan}\; \Gamma_{gw}}}\begin{matrix}{\left. \mspace{79mu} {{C^{''}C} = {{C^{\prime}C} - {C^{\prime}K}}} \right)\sin \; \Gamma_{gw}} \\{= {{L_{g}\left( {{\sin \; \phi_{nw}} - {\cos \; \phi_{nw}\sin \; {\psi_{gw}/\tan}\; \Gamma_{gw}}} \right)}\sin \; \Gamma_{gw}}}\end{matrix}\mspace{79mu} {{D^{\prime}D} = {B^{\prime}B}}\mspace{79mu} {{P_{w}E} = {P_{w}A}}\mspace{79mu} {{E^{\prime}E} = {C^{''}C}}\begin{matrix}{\mspace{79mu} {{\sin \; \psi_{b\; 2w}} = {E^{\prime}{E/P_{w}}E}}} \\{= {C^{''}{C/L_{g}}}} \\{= {\left( {{\sin \; \phi_{nw}} - {\cos \; \phi_{nw}\sin \; {\psi_{gw}/\tan}\; \Gamma_{gw}}} \right)\sin \; \Gamma_{gw}}}\end{matrix}}} & (17) \\{\mspace{79mu} {{{{\tan \; \eta_{x\; 2w}} = {{C^{\prime}{C/P_{w}}C^{\prime}} = {\tan \; {\phi_{nw}/\sin}\; \psi_{gw}}}}\begin{matrix}{\mspace{79mu} {{P_{w}C} = \left( {{P_{w}{C^{\prime}}^{2}} + {C^{\prime}C^{2}}} \right)^{1/2}}} \\{= {L_{g} \times \left\{ {\left( {\cos \; \phi_{nw}\sin \; \psi_{gw}} \right)^{2} + \left( {\sin \; \phi_{nw}} \right)^{2}} \right\}^{1/2}}}\end{matrix}\mspace{79mu} {{P_{w}C^{''}} = {{P_{w}C\; \cos \left\{ {\eta_{x\; 2w} - \left( {{\Pi/2} - \Gamma_{gw}} \right)} \right\}} = {P_{w}C\; {\sin \left( {\eta_{{xw}\; 2} + \Gamma_{gw}} \right)}}}}}\begin{matrix}{{\tan \left( {\chi_{2w} - ɛ_{2w}} \right)} = {D^{\prime}{D/P_{w}}C^{''}}} \\{= {\cos \; \phi_{nw}\cos \; {\psi_{gw}/\begin{bmatrix}{\left\{ {\left( {\cos \; \phi_{nw}\sin \; \psi_{gw}} \right)^{2} + \left( {\sin \; \phi_{nw}} \right)^{2}} \right\}^{1/2} \times} \\{\sin \left( {\eta_{x\; 2w} + \Gamma_{gw}} \right)}\end{bmatrix}}}}\end{matrix}}} & (18) \\{\mspace{79mu} {\phi_{2w} = {{\Pi/2} - \chi_{2w}}}} & \;\end{matrix}$

When g_(w) (φ_(2w), ψ_(b2w): C₂) is transformed from the coordinatesystem C₂ to the coordinate system C_(s), g_(w) (ψ₀, φ_(n0): C_(s)) canbe represented as follows:

sin φ_(n0)=cos ψ_(b2w) sin φ_(2w) cos Γ_(s)+sin ψ_(b2w) sin Γ_(s)

tan ψ₀=tan φ_(2w) sin Γ_(s)−tan ψ_(b2w) cos Γ_(s)/cos φ_(2w)  (19)

With the expressions (17), (18), and (19), g_(w) (ψ_(gw), φ_(nw)) can berepresented by g_(w) (ψ₀, φ_(n0): C_(s)).4.2B Reference Point P_(w) Based on R_(2w), β_(w), ψ_(rw)

As described above at the beginning of section 4.2A, the pitch cones ofthe pinion and the gear do not have to contact at the reference pointP_(w). In this section, a method is described in which the referencepoint P_(w) is determined on the coordinate system C_(s) without the useof the pitch cone, and by setting the gear reference radius R_(2w), aphase angle β_(w), and a spiral angle ψ_(rw) of the reference point.

The reference point P_(w) is set in the coordinate system C_(s) asfollows:

P _(w)(u _(cw) ,v _(cw) ,z _(cw) :C _(s))

When P_(w) is represented with the circle radius r_(w) of the relativevelocity, and an angle from the u_(c) axis β_(w), in a form ofP_(w)(r_(w), β_(w), z_(cw): C_(s)),

u _(cw) =r _(w) cos β_(w)

v _(cw) =r _(w) sin β_(w)

In addition, as the phase angle β_(w) of the reference point and thespiral angle ψ_(rw) are set based on expression (12) which represents arelationship between a radius r_(w) around the instantaneous axis of thereference point P_(w) and the inclination angle ψ_(rw) of the relativevelocity,

r _(w) =E tan ψ_(rw)×sin(Σ−Γ_(s))sin Γ_(s)/sin Σ

u_(cw) and v_(cw) are determined accordingly.

Next, P_(w)(u_(cw), v_(cw), z_(cw): C_(s)) is converted to thecoordinate system C₂ of rotation axis of the second gear. This isalready described as expression (13).

u _(2cw) =−u _(cw) cos Γ_(s) +z _(cw) sin Γ_(s)

v _(2cw) =v _(cw) −v _(cs2)

z _(2cw) =−u _(cw) sin Γ_(s) −z _(cw) cos Γ_(s)  (13a)

Here, as described in section 4.1, v_(cs2)=E tan Γ_(s)/{tan(Σ−Γ_(s))+tanΓ_(s)}. In addition, there is an expression in expression (13)describing:

R _(2w) ² =u _(2cw) ² +v _(2cw) ²  (13b)

Thus, by setting the gear reference radius R_(2w), z_(cw) is determinedbased on expressions (13a) and (13b), and the coordinate of thereference point P_(w) in the coordinate system C_(s) is calculated.

Once the design reference point P_(w) is determined, the pinionreference circle radius R_(1w) can also be calculated based onexpression (13).

Because the pitch generating line L_(pw) passing at the design referencepoint P_(w) is determined, the pitch hyperboloid can be determined.Alternatively, it is also possible to determine a design reference conein which the gear cone angle Γ_(gw) is approximated to be a value aroundΓ_(s), and the pinion cone angle γ_(pw) is approximated by Σ−Γ_(gw).Although the reference cones share the design reference point P_(w), thereference cones are not in contact with each other. The tip cone anglecan be determined similarly to as in section 4.2A.3.

A contact normal g_(w) is set as g_(w) (ψ_(rw), φ_(nrw); C_(rs)) asshown in FIG. 10. The variable φ_(nrw) represents an angle, on the planeS_(nw), between an intersecting line between the plane S_(nw) and theplane S_(pw) and the contact normal g_(w). The contact normal g_(w) canbe converted to g_(w) (ψ₀, φ_(n0); C_(s)) as will be described later.Because ψ_(pw) and ψ_(gw) can be determined based on expression (15),the contact normal g_(w) can be set as g_(w) (ψ_(pw), φ_(nw); S_(nw))similar to section 4.2A.4.

Conversion of the contact normal from the coordinate system C_(rs) tothe coordinate system C_(s) will now be described.

(1) A contact normal g_(w) (ψ_(rw), φ_(nrw); C_(rs)) is set.

(2) When the displacement on the contact normal g_(w) is L_(g), theaxial direction components of the displacement L_(g) on the coordinatesystem C_(rs) are:

L _(urs) =−L _(g) sin φ_(nrw)

L _(vrs) =L _(g) cos φ_(nrw)·cos ψ_(rw)

L _(zrs) =L _(g) cos φ_(nrw)·sin ψ_(rw)

(3) The axial direction components of the coordinate system C_(s) arerepresented with (L_(urs), L_(vrs), L_(zrs)) as:

L _(uc) =L _(urs)·cos β_(w) −L _(vrs)·sin β_(w)

L _(vc) =L _(urs)·sin β_(w) +L _(vrs)·cos β_(w)

L _(zc) =L _(zrs)

(4) Based on these expressions,

$\begin{matrix}{L_{uc} = {{{- \left( {L_{g}\sin \; \phi_{nrw}} \right)}\cos \; \beta_{w}} - {\left( {L_{g}\cos \; {\phi_{nrw} \cdot \cos}\; \psi_{rw}} \right)\sin \; \beta_{w}}}} \\{= {- {L_{g}\left( {{\sin \; {\phi_{nrw} \cdot \cos}\; \beta_{w}} + {\cos \; {\phi_{nrw} \cdot \cos}\; {\psi_{rw} \cdot \sin}\; \beta_{w}}} \right)}}}\end{matrix}$ $\begin{matrix}{L_{vc} = {{{- \left( {L_{g}\sin \; \phi_{nrw}} \right)}\sin \; \beta_{w}} + {\left( {L_{g}\cos \; {\phi_{nrw} \cdot \cos}\; \psi_{rw}} \right)\cos \; \beta_{w}}}} \\{= {L_{g}\left( {{{- \sin}\; {\phi_{nrw} \cdot \sin}\; \beta_{w}} + {\cos \; {\phi_{nrw} \cdot \cos}\; {\psi_{rw} \cdot \cos}\; \beta_{w}}} \right)}}\end{matrix}$

(5) From FIG. 6, the contact normal g_(w) (ψ₀, φ_(n0); C_(s)) is:

$\begin{matrix}{{\tan \; \psi_{0}} = {L_{zc}/L_{vc}}} \\{= {\cos \; {\phi_{nrw} \cdot \sin}\; {\psi_{rw}/\left( {{{- \sin}\; {\phi_{nrw} \cdot \sin}\; \beta_{w}} + {\cos \; {\phi_{nrw} \cdot \cos}\; {\psi_{rw} \cdot \cos}\; \beta_{w}}} \right)}}}\end{matrix}$ $\begin{matrix}{\mspace{79mu} {{\sin \; \phi_{n\; 0}} = {{- L_{uc}}/L_{g}}}} \\{= {{\sin \; {\phi_{nrw} \cdot \cos}\; \beta_{w}} + {\cos \; {\phi_{nrw} \cdot \cos}\; {\psi_{rw} \cdot \sin}\; \beta_{w}}}}\end{matrix}$

(6) From FIG. 11, the contact normal g_(w) (φ_(s0), ψ_(sw0); C_(s)) is:

tan φ_(s0) =−L _(uc) /L _(vc)=(sin φ_(nrw)·cos β_(w)+cos φ_(nrw)·cosψ_(rw)·sin β_(w))/(−sin φ_(nrw)·sin β_(w)+cos φ_(nrw)·cos ψ_(rw)·cosβ_(w))

sin ψ_(sw0) =L _(sc) /L _(g)=cos φ_(nrw)·cos ψ_(rw)

The simplest practical method is a method in which the design referencepoint P_(w) is determined with β_(w) set as β_(w)=0, and reference conesare selected in which the gear cone angle is around Γ_(gw)=Γ_(s) and thepinion cone angle is around γ_(pw)=Σ−Γ_(gw). In this method, becauseβ_(w)=0, the contact normal g_(w) is directly set as g_(w) (ψ₀, φ_(n0);C_(s)).

4.3 Tooth Trace Contact Ratio 4.3.1 General Equation of Tooth TraceContact Ratio

An contact ratio m_(f) along L_(pw) and an contact ratio m_(fcone) alonga direction of a line of intersection (P_(w)P_(gcone) in FIG. 13)between an arbitrary cone surface and S_(w0) are calculated with anarbitrary point P_(w) on g_(w)=g₀ as a reference. The other contactratios m_(z) and m_(s) are similarly determined.

Because the contact normal g_(w) is represented in the coordinate systemC_(s) with g_(w)=g₀ (ψ₀, φ_(n0): C_(s)) the point P_(w)(u_(2cw),v_(2cw), z_(2cw): C₂) represented in the coordinate system C₂ isconverted into the point P_(w)(q_(2cw), −R_(b2w), z_(2cw): C_(q2)) onthe coordinate system C_(q2) in the following manner:

$\begin{matrix}{{q_{2{cw}} = {{u_{2{cw}}\cos \; \chi_{20}} + {v_{2{cw}}\sin \; \chi_{20}}}}\begin{matrix}{R_{b\; 2w} = {{u_{2{cw}}\sin \; \chi_{20}} - {v_{2{cw}}\cos \; \chi_{20}}}} \\{= {R_{2w}{\cos \left( {\phi_{20} + ɛ_{2w}} \right)}}}\end{matrix}{\chi_{20} = {{\Pi/2} - \phi_{20}}}{{\tan \; ɛ_{2w}} = {v_{2{cw}}/u_{2{cw}}}}{R_{2w} = \left( {u_{2{cw}}^{2} + v_{2{cw}}^{2}} \right)^{1/2}}} & (20)\end{matrix}$

The inclination angle g₀ (φ₂₀, ψ_(b20): C₂) of the contact normalg_(w)=g₀, the inclination angle φ_(a0) of the surface of action S_(w0),and the inclination angle ψ_(sw0) of g₀ (=P₀P_(Gswn)) on S_(w0) (FIGS.12 and 13) are determined in the following manner:

(a) for Cylindrical Gears, Crossed Helical Gears, and Worm Gears

tan φ₂₀=tan φ_(n0) cos(Γ_(s)−ψ₀)

sin ψ_(b20)=sin φ_(n0) sin(Γ_(s)−ψ₀)

tan φ_(s0)=tan φ_(no) cos ψ₀

tan ψ_(sw0)=tan ψ_(s) sin φ_(s0)

or sin ψ_(sw0)=sin φ_(n0) sin ψ₀  (20a)

(b) for Bevel Gears and Hypoid Gears

tan φ₂₀=tan φ_(n0) cos Γ_(s)/cos ψ₀+tan ψ₀ sin Γ_(s)

sin ψ_(b20)=sin φ_(n0)=sin Γ_(s)−cos φ_(n0) sin ψ₀ cos Γ_(s)

tan φ_(a0)=tan φ_(n0)/cos ψ₀

tan ψ_(sw0)=tan ψ₀ cos φ_(a0)  (20b)

The derivation of φ_(s0) and φ_(sw0) are detailed in, for example,Papers of Japan Society of Mechanical Engineers, Part C, Vol. 70, No.692, c2004-4, Third Report of Design Theory of Power Transmission Gears.

In the following, a calculation is described in the case where the pathof contact coincides with the contact normal g_(w)=g₀. If it is assumedthat with every rotation of one pitch P_(w) moves to P_(g), and thetangential plane W translates to W_(g), the movement distance P_(w)P_(g)can be represented as follows (FIG. 11):

P _(w) P _(g) =P _(gw) =R _(b2w)(2θ_(2p))cos ψ_(b20)  (21)

where P_(gw) represents one pitch on g₀ and 2θ_(2p) represents anangular pitch of the ring gear.

When the intersection between L_(pw) and W_(g) is P_(1w), one pitchP_(fw)=P_(1w)P_(w) on the tooth trace L_(pw) is:

P _(fw) =P _(gw)/sin ψ_(sw0)  (22)

The relationship between the internal and external circle radii of thering gear and the face width of the ring gear is:

R _(2t) =R _(2h) −F _(g)/sin Γ_(gw)

where R_(2t) and R_(2h) represent internal and external circle radii ofthe ring gear, respectively, F_(g) represents a gear face width on thepitch cone element, and Γ_(gw) represents a pitch cone angle.

Because the effective length F_(1wp) of the tooth trace is a length ofthe pitch generating line L_(pw) which is cut by the internal andexternal circles of the ring gear:

F _(1wp)={(R _(2h) ² −v _(2pw) ²)^(1/2)−(R _(2t) ² −v _(2pw)²)^(1/2)}/sin Γ_(s)  (23)

Therefore, the general equation for the tooth trace contact ratio m_(f)would be:

m _(f) =F _(1wp) /P _(fw)  (24)

4.3.2 for Cylindrical Gear (FIG. 12)

The pitch generating line L_(pw) coincides with the instantaneous axis(Γ_(s)=0), and P_(w) may be anywhere on L_(pw). Normally, P_(w) is takenat the origin of the coordinate system C_(s), and, thus, P_(w) (u_(cw),v_(cw), z_(cw): C_(s)) and the contact normal g_(w)=g₀ (ψ₀, φ_(n0):C_(s)) can be simplified as follows, based on expressions (20) and(20a):

P _(w)(0,0,0:C _(s)),P _(w)(0,−v _(cs2) ,O:C ₂)

P ₀(q _(2pw) =−v _(cs2) sin χ₂₀ ,−R _(b2w) =−v _(cs2) cos χ₂₀,0:C _(q2))

φ₂₀=φ_(s0),ψ_(b20)=−ψ_(sw0)

tan ψ_(b20)=−tan ψ_(sw0)=−tan ψ₀ sin φ₂₀

In other words, the plane S_(w0) and the plane of action G₂₀ coincidewith each other. It should be noted, however, that the planes are viewedfrom opposite directions from each other.

These values can be substituted into expressions (21) and (22) todetermine the tooth trace contact ratio m_(f) with the tooth tracedirection pitch P_(fw) and expression (24):

$\begin{matrix}{{P_{gw} = {{R_{b\; 2w}\left( {2\theta_{2p}} \right)}\cos \; \psi_{b\; 20}}}{P_{fw} = {{{{P_{gw}/\sin}\; \psi_{{sw}\; 0}}} = {{{{R_{{b2}\; w}\left( {2\theta_{2p}} \right)}/\tan}\; \psi_{b\; 20}}}}}\begin{matrix}{m_{f} = {F_{1{wp}}/P_{fw}}} \\{= {F\; \tan \; {\psi_{0}/{R_{2w}\left( {2\theta_{2p}} \right)}}}} \\{= {F\; \tan \; {\psi_{0}/p}}}\end{matrix}} & (25)\end{matrix}$

where R_(2w)=R_(b2w)/sin φ₂₀ represents a radius of a ring gearreference cylinder, p=R_(2w) (2θ_(2p)) represents a circular pitch, andF=F_(1wp) represents the effective face width.

The expression (25) is a calculation equation of the tooth trace contactratio of the cylindrical gear of the related art, which is determinedwith only p, F, and ψ₀ and which does not depend on φ_(n0). This is aspecial case, which is only true when Γ_(s)=0, and the plane S_(w0) andthe plane of action G₂₀ coincide with each other.

4.3.3 for Bevel Gears and Hypoid Gears

For the bevel gears and the hypoid gears, the plane S_(w0) does notcoincide with G₂₀ (S_(w0)≠G₂₀), and thus the tooth trace contact ratiom_(f) depends on φ_(n0), and would differ between the drive-side and thecoast-side. Therefore, the tooth trace contact ratio m_(f) of the bevelgear or the hypoid gear cannot be determined with the currently usedexpression (25). In order to check the cases where the currently usedexpression (25) can hold, the following conditions (a), (b), and (c) areassumed:

(a) the gear is a bevel gear; therefore, the pitch generating lineL_(pw) coincides with the instantaneous axis and the design referencepoint is P_(w)(0, 0, z_(cw): C_(s));

(b) the gear is a crown gear; therefore, Γ_(s)=π/2; and

(c) the path of contact is on the pitch plane; therefore, φ_(n0)=0.

The expressions (20), and (20b)-(24) can be transformed to yield:

φ₂₀=ψ₀,ψ_(b20)=0,φ_(s0)=0,ψ_(sw0)=ψ0

R _(b2w) =R _(2w) cos φ₂₀ =R _(2w) cos ψ₀

P _(gw) =R _(b2w)(2θ_(2p))cos ψ_(b20) =R _(2w)(2θ₂₉)cos ψ₀

P _(fw) =|P _(gw)/sin ψ_(sw0) |=|R _(2w)(2θ_(2p))/tan ψ₀|

m _(f) =F _(1wp) /P _(fw) =F tan ψ₀ /R _(2w)(2θ_(2p))=F tan ψ₀ /p  (26)

The expression (26) is identical to expression (25). In other words, thecurrently used expression (25) holds in bevel gears which satisfy theabove-described conditions (a), (b), and (c). Therefore,

(1) strictly, the expression cannot be applied to normal bevel gearshaving Γ_(s) different from π/2 (Γ_(s)≠π/2) and φ_(n0) different from 0(φ_(n0)≠0); and

(2) in a hypoid gear (E≠0), the crown gear does not exist and ε_(2w)differs from 0 (ε_(2w)≠0).

For these reasons, the tooth trace contact ratios of general bevel gearsand hypoid gears must be determined with the general expression (24),not the expression (26).

4.4 Calculation Method of Contact Ratio m_(fcone) Along Line ofIntersection of Gear Pitch Cone and Surface of Action S_(w0)

The tooth trace contact ratios of the hypoid gear (Gleason method) iscalculated based on the expression (26), with an assumption of a virtualspiral bevel gear of ψ₀=(ψ_(pw)+ψ_(gw))/2 (FIG. 9), and this value isassumed to be sufficiently practical. However, there is no theoreticalbasis for this assumption. In reality, because the line of intersectionof the gear pitch cone and the tooth surface is assumed to be the toothtrace curve, the contact ratio is more properly calculated along theline of intersection of the gear pitch cone and the surface of actionS_(w0) in the static coordinate system. In the following, the contactratio m_(fcone) of the hypoid gear is calculated from this viewpoint.

FIG. 13 shows a line of intersection P_(w)P_(gcone) with an arbitrarycone surface which passes through P_(w) on the surface of action S_(w0).Because P_(w)P_(gcone) is a cone curve, it is not a straight line in astrict sense, but P_(w)P_(gcone) is assumed to be a straight line herebecause the difference is small. When the line of intersection betweenthe surface of action S_(w0) and the plane v_(2c)=0 is P_(ssw)P_(gswn)the line of intersection P_(ssw)P_(gswn) and an arbitrary cone surfacepassing through P_(w) have an intersection P_(gcone), which is expressedin the following manner:

P _(gcone)(u _(cgcone) ,v _(cs2) ,z _(cgcone) :C _(s))

P _(gcone)(u _(2cgcone),0,z _(c2gcone) :C ₂)

where

u_(cgcone)=u_(cw)+(v_(cw)−v_(cs2))tan Γ_(s0)

z_(cgcone)={(v_(cs2)−v_(cw))/cos φ_(s0)} tan ψ_(gcone)+z_(cw)

u_(2cgcone)=−u_(cgcone) cos Γ_(s)+z_(cgcone) sin Γ_(s)

z_(2cgcone)=−u_(cgcone) sin Γ_(s)−z_(cgcone) cos Γ_(s)

ψ_(gcone) represents an inclination angle of P_(w) P_(gcone) fromP₀P_(ssw) on S_(w0).

Because P_(gcone) is a point on a cone surface of a cone angle Γ_(gcone)passing through P_(w), the following relationship holds.

u _(2cgcone) −R _(2w)=−(z _(2cgcone) −z _(2cw))tan Γ_(gcone)  (27)

When a cone angle Γ_(gcone) is set, ψ_(gcone) can be determined throughexpression (27). Therefore, one pitch P_(cone) along P_(w)P_(gcone) is:

P _(cone) =P _(gw)/cos(ψ_(gcone)−ψ_(sw0))  (28)

The contact length F_(1wpcone) along P_(w)P_(gcone) can be determined inthe following manner.

In FIG. 11, if an intersection between P_(w)P_(gcone) and L_(pw0) isP_(ws) (u_(cws), 0, z_(cws): C_(s))

u _(cws) =u _(cw) +v _(cw) tan φ_(s0)

z _(cws) =z _(cw)−(v _(cw)/cos φ_(s0))tan ψ_(gcone)

If an arbitrary point on the straight line P_(w)P_(gcone) is set asQ(u_(cq), v_(cq), z_(cq): C_(s)) (FIG. 11), u_(cq) and v_(cq) can berepresented as functions of z_(cq):

v _(cq)={(z _(cq) −z _(cws))/tan ψ_(gcone)} cos φ_(a0)

u _(cq) =u _(cws) −v _(cq) tan φ_(s0)

If the point Q is represented in the coordinate system C₂ usingexpression (13), to result in Q (u_(2cq), v_(2cq), z_(2cq): C₂), theradius R_(2cq) of the point Q is:

u _(2cq) =−u _(cq) cos Γ_(s) +z _(cq) sin Γ_(s)

v _(2cq) =v _(cq) −v _(cs2)

R _(2cq)=√(u _(2cq) ² +v _(2cq) ²)

If the values of z_(cq) where R_(2cq)=R_(2h) and R_(2cq)=R_(2t) arez_(cqh) and z_(cqt), the contact length F_(1wpcone) is:

F _(1wpcone)=(z _(cqh) −z _(cqt))/sin ψ_(gcone)  (29)

Therefore, the contact ratio m_(fcone) along P_(w)P_(gcone) is:

m _(fcone) =F _(1wpcone) /P _(cone)  (30)

The value of m_(fcone) where ψ_(gcone)→π/2 (expression (30)) is thetooth trace contact ratio m_(f) (expression (24)).

5. Examples

Table 1 shows specifications of a hypoid gear designed through theGleason method. The pitch cone is selected such that the radius ofcurvature of the tooth trace=cutter radius R_(c)=3.75″. In thefollowing, according to the above-described method, the appropriatenessof the present embodiment will be shown with a test result by:

(1) first, designing a hypoid gear having the same pitch cone and thesame contact normal as Gleason's and calculating the contact ratiom_(fcone) in the direction of the line of intersection of the pitch coneand the surface of action, and

(2) then, designing a hypoid gear with the same ring gear referencecircle radius R_(2w), the same pinion spiral angle ψ_(pw), and the sameinclination angle φ_(nw) of the contact normal, in which the tooth tracecontact ratio on the drive-side and the coast-side are approximatelyequal to each other.

5.1 Uniform Coordinate Systems C_(s), C₁, and C₂, Reference Point P_(w)and Pitch Generating Line L_(pw)

When values of a shaft angle Σ=90°, an offset E=28 mm, and a gear ratioi₀=47/19 are set, the intersection C_(s) between the instantaneous axisand the line of centers and the inclination angle Γ_(s) of theinstantaneous axis are determined in the following manner with respectto the coordinate systems C₁ and C₂:

-   -   C_(s)(0, 24.067, 0: C₂), C_(s)(0, −3.993, 0: C₁), Γ_(s)=67.989°

Based on Table 1, when values of a ring gear reference circle radiusR_(2w)=89.255 mm, a pinion spiral angle ψ_(pw)=46.988°, and a ring gearpitch cone angle Γ_(gw)=62.784° are set, the system of equations basedon expressions (13), (14), and (15) would have a solution:

-   -   r_(w)=9.536 mm, β_(w)=11.10°, z_(cw)=97.021

Therefore, the pitch point P_(w) is:

-   -   P_(w)(9.358, 1.836, 97.021: C_(s))

The pitch generating line L_(pw) is determined on the coordinate systemC_(s) as a straight line passing through the reference point P_(w) andparallel to the instantaneous axis (Γ_(s)=67.989°).

In the following calculations, the internal and external circle radii ofthe ring gear, R_(2t)=73.87 and R_(2h)=105 are set to be constants.

5.2 Contact Ratio m_(fconeD) of Tooth Surface D (Represented with Indexof D) with Contact Normal g_(wD)

Based on Table 1, when g_(wD) is set with g_(wD)(ψ_(gw)=30.859°,φ_(nwD)=15°), g_(wD) can be converted into coordinate systems C_(s) andC₂ with expressions (17), (18), and (19), to yield:

-   -   g_(wD) (φ_(20D)=48.41°, ψ_(b20D)=0.20°: C₂)    -   g_(wD)(ψ_(0D)=46.19°, φ_(n0D)=16.48°: C_(s))

The surface of action S_(wD) can be determined on the coordinate systemC_(s) by the pitch generating line L_(pw) and g_(wD). In addition, theintersection P_(0d) between g_(wD) and the plane S_(H) and the radiusR_(20D) around the gear axis are, based on expression (16):

-   -   P_(0D)(10.142, 0, 95.107: C_(s)), R_(20D)=87.739 mm

The contact ratio m_(fconeD) in the direction of the line ofintersection between the pitch cone and the surface of action isdetermined in the following manner.

The inclination angle φ_(s0D) of the surface of action S_(wD), theinclination angle ψ_(sw0D) of g_(wD) on S_(wD), and one pitch P_(gwD) ong_(wD) are determined, based on expressions (20), (20b), and (21), as:

-   -   φ_(s0D)=23.13°, ψ_(sw0D)=43.79°, P_(gwD)=9.894        (1) When Γ_(gw)=Γ_(gcone)=62.784° is set, based on expressions        (27)-(30),    -   ψ_(gcone63D)=74.98°, P_(cone63D)=20.56,    -   F_(1wpcone63D)=34.98, m_(fcone63D)=1.701.        (2) When Γ_(gcone)=Γ_(s)=67.989° is set, similarly,    -   ψ_(gcone68D)=−89.99°, P_(cone68D)=14.30,    -   F_(1wpcone68D)=34 70, m_(fcone68D)=2.427.        (3) When Γ_(gcone)=72.0° is set, similarly,    -   ψ_(gcone72D)=78.88°, P_(cone72D)=12.09,    -   F_(1wpcone72D)=36.15, m_(fcone72D)=2.989.        5.3 Contact Ratio m_(fconeC) of Tooth Surface C (Represented        with Index C) with Contact Normal g_(wC)

When g_(wC)(ψ_(gw)=30.859°, φ_(nwC)=−27.5° is set, similar to the toothsurface D,

-   -   g_(wC)(φ_(20C)=28.68°, ψ_(b20C)=−38.22°: C₂)    -   g_(wC)(ψ_(0C)=40.15°, φ_(n0C)=−25.61°: C_(s))    -   P_(0C)(8.206, 0, 95.473: C_(s)), R_(20C)=88.763 mm

The inclination angle φ_(s0C) of the surface of action S_(wC), theinclination angle ψ_(sw0C) of g_(wC) on S_(wc), and one pitch P_(gwC) ong_(wC) are, based on expressions (20), (20b), and (21):

-   -   φ_(s0C)=−32.10°, ψ_(sw0C)=35.55°, P_(gwC)=9.086        (1) When Γ_(gw)=Γ_(gcone)=62.784° is set, based on expressions        (27)-(30),    -   ψ_(gcone63C)=81.08°, P_(cone63C)=12.971,    -   F_(1wpcone63C)=37.86, m_(fcone63C)=2.919.        (2) When Γ_(gcone)=Γ_(s)=67.989° is set, similarly,    -   ψ_(gcone68C)=89.99°, P_(cone68C)=15.628,    -   F_(1wpcone68C)=34.70, m_(fcone68C)=2.220.        (3) When Γ_(gcone)=72° is set, similarly,    -   ψ_(gcone72C)=82.92°, P_(cone72C)=19.061,    -   F_(1wpcone72C)=33.09, m_(fcone72C)=1.736.

According to the Gleason design method, becauseΓ_(gw)=Γ_(gcone)=62.784°, the contact ratio along the line ofintersection between the pitch cone and the surface of action arem_(fcone63D)=1.70 and m_(fcone63C)=2.92, which is very disadvantageousfor the tooth surface D. This calculation result can be considered to beexplaining the test result of FIG. 16.

In addition, when the ring gear cone angle Γ_(gcone)=Γ_(s)=67.989°,ψ_(gcone)=−89.99° in both the drive-side and the coast-side. Thus, theline of intersection between the cone surface and the surface of actioncoincides with the pitch generating line L_(pw), the tooth trace contactratio of the present invention is achieved, and the contact ratio isapproximately equal between the drive-side and the coast-side. Becauseof this, as shown in FIG. 17, a virtual pitch cone C_(pv) passingthrough a reference point P_(w) determined on the pitch cone angle ofΓ_(gw)=62.784° and having the cone angle of Γ_(gcone)=67.989° and apinion virtual cone (not shown) having the cone angleγ_(pcone)=Σ−Γ_(gcone)=22.02° can be defined, and the addendum, addendumangle, dedendum, and dedendum angle of the hypoid gear can be determinedaccording to the following standard expressions of gear design, withreference to the virtual pitch cone. In the gear determined as describedabove, the tooth trace contact ratio of the present embodiment can berealized along the virtual pitch cone angle.

α_(g)=Σδ_(t) ×a _(g)/(a _(g) +a _(p))  (31)

a _(g) +a _(p) =h _(k)(action tooth size)  (32)

where Σδ_(t) represents a sum of the ring gear addendum angle and thering gear dedendum angle (which changes depending on the tapered toothdepth), α_(g) represents the ring gear addendum angle, a_(g) representsthe ring gear addendum, and a_(p) represents the pinion addendum.

The virtual pitch cones C_(pv) of the ring gear and the pinion definedhere do not contact each other, although the cones pass through thereference point P_(w1).

The addendum and the addendum angle are defined as shown in FIGS. 18 and19. More specifically, the addendum angle α_(g) of a ring gear 100 is adifference between cone angles of a pitch cone 102 and a cone 104generated by the tooth tip of the ring gear, and the dedendum angleβ_(g) is similarly a difference between cone angles of the pitch cone102 and a cone 106 generated by the tooth root of the ring gear. Anaddendum a_(g) of the ring gear 100 is a distance between the designreference point P_(w) and the gear tooth tip 104 on a straight linewhich passes through the design reference point P_(w) and which isorthogonal to the pitch cone 102, and the dedendum b_(g) is similarly adistance between the design reference point P_(w) and the tooth root 106on the above-described straight line. Similar definitions apply for apinion 110. By changing the pitch cone angle such that, for example,Γ_(gw)=72°>Γ_(s), it is possible to design the tooth trace contact ratioto be larger on the tooth surface D and smaller on the tooth surface C.Conversely, by changing the pitch cone angle such that, for example,Γ_(gw)=62.784°<Γ_(s), the tooth trace contact ratio would be smaller onthe tooth surface D and larger on the tooth surface C.

A design method by the virtual pitch cone C_(pv) will now beadditionally described. FIG. 17 shows a pitch cone C_(p1) having a coneangle Γ_(gw)=62.784° according to the Gleason design method and thedesign reference point P_(w1). As described above, in the Gleason designmethod, the drive-side tooth surface is disadvantageous in view of thecontact ratio. When, on the other hand, the gear is designed with apitch cone C_(p2) having a cone angle Γ_(gw)=67.989°, the contact ratiocan be improved. The design reference point P_(w2) in this case is apoint of contact between the pitch cones of the ring gear and thepinion. In other words, the design reference point is changed from thereference point P_(w1) determined based on the Gleason design method tothe reference point P_(w2) so that the design reference point is at thepoint of contact between pitch cones of the ring gear and the pinion.

As already described, if the surface of action intersects the conesurface having the cone angle of Γ_(gw)=67.989° over the entire facewidth, the above-described tooth trace contact ratio can be realized. Inother words, in FIG. 17, when the element of the cone (virtual pitchcone) passing through the design reference point P_(w1) and having thecone angle of 67.989° exists in the gear tooth surface in the face widthof the ring gear, the above-described contact ratio can be realized. Inorder to realize this, a method may be considered in which the addendumangle and the dedendum angle are changed according to the currentmethod. However, this method cannot be realized due to the followingreason.

In order for the cone surface having the cone angle of 67.989°(approximately 68°) and the surface of action to intersect over theentire face width without a change in the pitch cone C_(p1), the ringgear addendum angle α_(g) may be increased so that the tip cone angleΓ_(f) is 68°. As shown in FIG. 20, by setting the gear addendum angleα_(g) to 5.216°, the tip cone angle Γ_(f)=68° is realized and theabove-described tooth trace contact ratio is achieved along the toothtip. However, if the tooth is designed according to the standardexpressions (31) and (32), almost no dedendum of the ring gear exists,and the pinion would consist mostly of the dedendum. In this case, thepinion would have negative addendum modification, sufficient effectivetooth surface cannot be formed, and the strength of the tooth of thepinion is reduced. Thus, such a configuration cannot be realized.

5.4 Hypoid Gear Specifications and Test Results when Γ_(gw) is SetΓ_(gw)=Γ_(s)=67.989°

Table 2 shows hypoid gear specifications when Γ_(gw) is setΓ_(gw)=Γ_(s)=67.989°. Compared to Table 1, identical ring gear referencecircle radius R_(2w)=89.255 mm and pinion spiral angle ψ_(pw)=46.988°are employed, and the ring gear pitch cone angle is changed fromΓ_(gw)=62.784° to 67.989°. As a result, P_(w) and Γ_(gw) differ as shownin FIG. 17 and, as will be described below, the other specifications arealso different. The pitch cone of the gear is in contact with the pitchcone of the pinion at the reference point P_(w).

-   -   Design reference point P_(w)(9.607, 0.825, 96.835: C_(s))    -   Pinion cone radius R_(1w)=45.449 mm    -   Ring gear pitch cone angle Γ_(gw)=67.989°    -   Pinion pitch cone angle γ_(p)=21.214°    -   Spiral angle on ring gear pitch plane ψ_(gw)=30.768°

With the pressure angles φ_(nwD) and φ_(nwC) identical to Table 1, ifg_(wD)(30.768°, 15°) and g_(wC)(30.768°, −27.5°) are set, theinclination angles would differ, in the static coordinate system C_(s),from g_(wD) and g_(wC) of Table 1:

-   -   g_(wD)(ψ_(0D)=45.86°, φ_(n0D)=19.43°: C_(s))    -   g_(wC)(ψ_(0c)=43.17°, φ_(n0C)=−22.99°: C_(s))

The inclination angles of g_(wD) and g_(wC) on the surface of action,and one pitch are:

-   -   φ_(s0D)=26.86°, ψ_(sw0D)=42.59°, P_(gwD)=9.903    -   φ_(s0C)=−30.19°, ψ_(sw0C)=39.04°, P_(gwC)=9.094

The tooth trace contact ratios are calculated in the following mannerbased on expressions (22), (23), and (24):

-   -   Drive-side: P_(fwD)=14.63, F_(1wpD)=34.70, m_(fD)=2.371    -   Coast-side: P_(fwc)=14.44, F_(1wpC)=34.70, m_(fC)=2.403

FIG. 21 shows a test result of the specifications of Table 2, and it canbe seen that, based on a comparison with FIG. 16, the transmission erroris approximately equal between the drive-side and the coast-side,corresponding to the tooth trace contact ratios.

5.5 Specifications of Hypoid Gear when β_(w)=0

Table 3 shows specifications of a hypoid gear when β_(w) is set to 0(β_(w)=0) in the method of determining the design reference point P_(w)based on R_(2w), β_(w), and ψ_(rw) described in section 4.2B.

6. Computer Aided Design System

In the above-described design of hypoid gears, the design is aided by acomputer aided system (CAD) shown in FIG. 22. The CAD system comprises acomputer 3 having a processor 1 and a memory 2, an inputting device 4,an outputting device 5, and an external storage device 6. In theexternal storage device 6, data is read and written from and to arecording medium. On the recording medium, a gear design program forexecuting the design method of the hypoid gear as described above isrecorded in advance, and the program is read from the recording mediumas necessary and executed by the computer.

The program can be briefly described as follows. First, a design requestvalue of the hypoid gear and values of variables for determining a toothsurface are acquired. A pitch cone angle Γ_(gcone) of one gear isprovisionally set and used along with the acquired values of thevariables, and an contact ratio m_(fconeD) of the drive-side toothsurface and an contact ratio m_(fconeC) of the coast-side tooth surfacebased on the newly defined tooth trace as described above arecalculated. The pitch cone angle Γ_(gcone) is changed and thecalculation is repeatedly executed so that these contact ratios becomepredetermined values. When the contact ratios of the tooth surfacesbecome predetermined values, the pitch cone angle at this point is setas a design value Γ_(gw), and the specifications of the hypoid gear arecalculated. The predetermined value of the contact ratio designates acertain range, and values in the range. Desirably, the range of thecontact ratio is greater than or equal to 2.0. The range may be changedbetween the drive-side and the coast-side. The initial value of thepitch cone angle Γ_(gcone) to be provisionally set is desirably set tothe inclination angle Γ_(s) of the instantaneous axis S.

Another program calculates the gear specifications by setting the pitchcone angle Γ_(gw), to the inclination angle Γ_(s) of the instantaneousaxis from the first place, and does not re-adjust the pitch cone angleaccording to the contact ratio. Because it is known that the contactratios of the tooth surfaces become approximately equal to each otherwhen the pitch cone angle Γ_(gw) is set to the inclination angle Γ_(s)of the instantaneous axis, such a program is sufficient as a simplemethod.

FIG. 23 shows a gear ratio and a tip cone angle (face angle) Γ_(f) of auniform tooth depth in which a tooth depth is constant along a facewidth direction, manufactured through a face hobbing designed by thecurrent method. The uniform tooth depth hypoid gear is a gear in whichboth the addendum angle α_(g) and the dedendum angle β_(g) of FIG. 19are 0°, and, consequently, the tip cone angle Γ_(f) is equal to thepitch cone angle Γ_(gw). The specifications of the tooth are determinedby setting addendum and dedendum in the uniform tooth depth hypoid gear.In a uniform tooth depth hypoid gear, as shown in FIG. 25, while acutter revolves around the center axis II of the ring gear 14, thecutter rotates with a cutter center cc as the center of rotation. Withthis motion, the edge of the cutter moves in an epicycloidal shape andthe tooth trace curve is also in an epicycloidal shape. In hypoid gearshaving a ratio r_(c)/D_(g0) between a cutter radius r_(c) and an outerdiameter D_(g0) of less than or equal to 0.52, a ratio E/D_(g0) betweenan offset E and the outer diameter D_(g0) of greater than or equal to0.111, and a gear ratio of greater than or equal to 2 and less than orequal to 5, as shown in FIG. 23, hypoid gears around the inclinationangle Γ_(s) of the instantaneous axis S are not designed. On the otherhand, with the design method of the cone angle according to the presentembodiment, hypoid gears having the tip cone angle Γ_(f) which is aroundΓ_(s) can be designed.

FIG. 24 shows a gear ratio and a tip cone angle (face angle) Γ_(f) of atapered tooth depth gear in which a tooth depth changes along a facewidth direction, manufactured through a face milling designed by thecurrent method. As shown in FIG. 19, the tip cone angle Γ_(f) is a sumof the pitch cone angle Γi_(gw) and the addendum angle α_(g), and is avalue determined by a sum of the ring gear addendum angle and the ringgear dedendum angle, the ring gear addendum, and the pinion addendum, asshown in expression (31). In a tapered tooth depth hypoid gear, as shownin FIG. 26, a radius of curvature of the tooth trace of the ring gear 14is equal to the cutter radius Γ_(c). In hypoid gears having a ratior_(c)/D_(g0) between a cutter radius r_(c) and an outer diameter D_(g0)of less than or equal to 0.52, a ratio E/D_(g0) between an offset E andthe outer diameter D_(g0) of greater than or equal to 0.111, and a gearratio of greater than or equal to 2 and less than or equal to 5, asshown in FIG. 24, hypoid gears of greater than or equal to theinclination angle Γ_(s) of the instantaneous axis S are not designed. Onthe other hand, the hypoid gear according to the present embodiment hasa tip cone angle of greater than or equal to Γ_(s) although thevariables are within the above-described range, as shown with reference“A” in FIG. 24. Therefore, specifications that depart from the currentmethod are designed.

TABLE 1 PINION RING GEAR SHAFT ANGLE Σ    90° OFFSET E 28 NUMBER OFTEETH N1, N2 19 47 INCLINATION ANGLE Γ_(s) OF 67.989° INSTANTANEOUS AXISCUTTER RADIUS R_(c) (RADIUS OF 3.75″ CURVATURE OF GEAR TOOTH TRACE)REFERENCE CIRCLE RADIUS 45.406 89.255 R1w, R2w PITCH CONE ANGLE γpw, Γgw26.291° 62.784° SPIRAL ANGLE ON PITCH PLANE 46.988° 30.858° φpw, φgw TIPCONE ANGLE 30.728° 63.713° INTERNAL AND EXTERNAL RADII 73.9, 105, (35)OF GEAR (FACE WIDTH) R2t, R2h (Fg) GEAR ADDENDUM 1.22 GEAR DEDENDUM 6.83GEAR WORKING DEPTH 7.15 CONTACT RATIO DRIVE-SIDE COAST-SIDE (GLEASONMETHOD) PRESSURE ANGLE φnw    15°  −27.5° TRANSVERSE CONTACT RATIO 1.130.78 TRACE CONTACT RATIO 2.45 2.45 (NEW CALCULATION METHOD (1.70) (2.92)mfcone)

TABLE 2 PINION RING GEAR SHAFT ANGLE Σ    90° OFFSET E 28 NUMBER OFTEETH N1, N2 19 47 INCLINATION ANGLE Γ_(s) OF 67.989° INSTANTANEOUS AXISCUTTER RADIUS R_(c) (RADIUS OF ARBITRARY CURVATURE OF GEAR TOOTH TRACE)REFERENCE CIRCLE RADIUS 45.449 89.255 R1w, R2w PITCH CONE ANGLEγpw, Γgw21.214° 67.989° SPIRAL ANGLE ON PITCH PLANE 46.988° 30.768° φpw, φgw TIPCONE ANGLE 25.267° 68.850° INTERNAL AND EXTERNAL RADII 73.9, 105 OF GEARR2t, R2h GEAR ADDENDUM 1.22 GEAR DEDENDUM 6.83 GEAR WORKING DEPTH 7.15CONTACT RATIO DRIVE-SIDE COAST-SIDE (NEW CALCULATION METHOD) PRESSUREANGLE φnw    15°  −27.5° TRANSVERSE CONTACT RATIO 1.05 0.85 ms TOOTHTRACE CONTACT RATIO 2.37 2.40 mf

TABLE 3 PINION RING GEAR SHAFT ANGLE Σ    90° OFFSET E 28 NUMBER OFTEETH N1, N2 19 47 INCLINATION ANGLE Γ_(s) OF 67.989° INSTANTANEOUS AXISCUTTER RADIUS R_(c) (RADIUS OF ARBITRARY CURVATURE OF GEAR TOOTH TRACE)DESIGN REFERENCE POINT P_(w) (9.73, 0, 96.64) REFERENCE CIRCLE RADIUS45.41 89.255 R_(1w), R_(2w) PITCH CONE ANGLE γ_(pw), Γ_(gw) 22° 68°SPIRAL ANGLE ψ_(rw) = ψ₀ 45° TIP CONE ANGLE 22° 68° INTERNAL ANDEXTERNAL RADII 73.9, 105 OF GEAR (FACE WIDTH) R_(2t), R_(2h) GEARADDENDUM 1.22 GEAR DEDENDUM 6.83 GEAR WORKING TOOTH DEPTH 7.15 CONTACTRATIO DRIVE-SIDE COAST-SIDE (NEW CALCULATION METHOD) PRESSURE ANGLEφ_(n0D), φ_(n0C) 18° −20°   TRANSVERSE CONTACT RATIO m_(s) 1.34 0.63TOOTH TRACE CONTACT RATIO 2.43 2.64 m_(f)

1. A hypoid gear comprising a pair of gears including a first gear and asecond gear, wherein a tip cone angle Γ_(f) of the second gear is set ata value in a predetermined range around an inclination angle Γ_(s) of aninstantaneous axis S which is an axis of a relative angular velocity ofthe first gear and the second gear with respect to a rotational axis ofthe second gear, and the predetermined range is a range in which each oftooth trace contact ratios of a drive-side tooth surface and acoast-side tooth surface is greater than or equal to 2.0.
 2. A hypoidgear comprising a pair of gears including a first gear and a secondgear, wherein a tip cone angle Γ_(f) of the second gear is set at avalue in a predetermined range around an inclination angle Γ_(s) of aninstantaneous axis S which is an axis of a relative angular velocity ofthe first gear and the second gear with respect to a rotational axis ofthe second gear.
 3. A hypoid gear comprising a pair of gears including afirst gear and a second gear, wherein a tip cone angle Γ_(f) of thesecond gear is set at a value which is greater than or equal to aninclination angle Γ_(s) of an instantaneous axis S which is an axis of arelative angular velocity of the first gear and the second gear withrespect to a rotational axis of the second gear.
 4. A uniform toothdepth hypoid gear comprising a pair of gears including a first gear anda second gear and in which a tooth depth is constant along a face widthdirection and a tooth trace shape is such that a radius of curvature ofa tooth trace is not constant, wherein a ratio r_(c)/D_(g0) between acutter radius r_(c) and an outer diameter D_(g0) is less than or equalto 0.52, a ratio E/D_(g0) between an offset E and the outer diameterD_(g0) is greater than or equal to 0.111, and a gear ratio is greaterthan or equal to 2 and less than or equal to 5, and a tip cone angleΓ_(f) of the second gear is set at a value which is within a range of±5° of an inclination angle Γ_(s) of an instantaneous axis S which is anaxis of a relative angular velocity of the first gear and the secondgear with respect to a rotational axis of the second gear.
 5. Theuniform tooth depth hypoid gear according to claim 4, wherein the toothtrace shape is an epicycloidal shape.
 6. An tapered tooth depth hypoidgear comprising a pair of gears including a first gear and a second gearand in which a tooth depth changes along a face width direction, whereina ratio r_(c)/D_(g0) between a radius of curvature r_(c) of a toothtrace of the second gear and an outer diameter D_(g0) is less than orequal to 0.52, a ratio E/D_(g0) between an offset E and the outerdiameter D_(g0) is greater than or equal to 0.111, and a gear ratio isgreater than or equal to 2 and less than or equal to 5, and a tip coneangle Γ_(f) of the second gear is set at a value which is greater thanor equal to an inclination angle Γ_(s), of an instantaneous axis S whichis an axis of a relative angular velocity of the first gear and thesecond gear with respect to a rotational axis of the second gear, andless than or equal to (Γ_(s)+5)°.